OFDM: From Foundations to 6G

1.1 From Fourier Series to the Discrete Fourier Transform

The journey toward the DFT begins with the observation that any sufficiently well-behaved periodic function can be decomposed into a weighted sum of harmonically related sinusoids — a fact first systematized by Joseph Fourier in 1822. Understanding the conceptual chain Fourier Series → Fourier Transform → DTFT → DFT shows that the DFT is not an ad-hoc engineering trick but the natural endpoint of discretizing and periodizing the continuous transform.

1.1.1 Fourier Series (continuous-time, periodic)

Let $x(t)$ be $T_0$-periodic. The Fourier Series (FS) representation is:

where the coefficients $c_k$ are obtained by the analysis integral:

The set $\{e^{j 2\pi k f_0 t}\}_{k \in \mathbb{Z}}$ forms a complete orthonormal basis for $L^2([0,T_0])$:

The frequency spectrum is discrete (harmonics at $kf_0$) but the time signal is continuous.

1.1.2 Continuous-Time Fourier Transform (CTFT)

When $T_0 \to \infty$ (the signal is no longer constrained to be periodic, or equivalently the fundamental frequency $f_0 \to df$ becomes infinitesimal), the harmonic sum becomes an integral — the Fourier Transform:

Both time and frequency are now continuous. Infinite support in time implies a continuous frequency spectrum. The spectrum is generally complex-valued; $|X(f)|$ is the amplitude spectrum and $\angle X(f)$ is the phase spectrum.

1.1.3 Discrete-Time Fourier Transform (DTFT)

Practical digital systems work with sequences $x[n]$ obtained by sampling $x(t)$ at interval $T_s$ (rate $f_s = 1/T_s$). The DTFT maps a discrete-time sequence to a continuous, periodic frequency function:

The DTFT is $2\pi$-periodic in $\omega$ (equivalently, $f_s$-periodic in Hz). It is still a continuous function of $\omega$ and requires an infinite sequence — not computable directly on a machine.

1.1.4 Arriving at the DFT

If we additionally limit the sequence to $N$ samples and evaluate the DTFT at $N$ equally-spaced frequencies $\omega_k = 2\pi k / N$ for $k = 0, 1, \ldots, N-1$, we obtain the Discrete Fourier Transform:

Both the time-domain sequence $x[n]$ and the frequency-domain sequence $X[k]$ are now finite and discrete — perfectly suited for digital computation. The DFT implicitly assumes both sequences are periodic with period $N$.

1.2 DFT and IDFT — Formal Definitions

Let $x[n]$, $n = 0, 1, \ldots, N-1$, be a finite sequence of (possibly complex) numbers. Define the N-th root of unity twiddle factor:

so that $W_N^{kn} = e^{-j 2\pi kn/N}$. The DFT pair is then written compactly as:

Physical interpretation of DFT bins:

• Bin $k=0$: DC component — $X[0] = \sum_n x[n]$ (sum of all samples).
• Bin $k$: complex amplitude of the sinusoid at normalized frequency $f_k = k/N$ cycles per sample, or in Hz: $f_k = k f_s / N$ where $f_s$ is the sample rate.
• Bins $k = N/2+1, \ldots, N-1$ correspond to negative frequencies $f = (k-N)/N$ (aliased by periodicity); for real $x[n]$, these are complex conjugates of bins $1$ through $N/2-1$.
• Bin $k = N/2$: the Nyquist frequency $f_s/2$.

1.3 Orthogonality of Complex Exponentials — Proof

The correctness of the IDFT formula — that it inverts the DFT exactly — rests entirely on the discrete orthogonality of the complex exponential basis functions. We now prove this fundamental identity.

Proof. Let $r = k - m$. Consider two cases:

Case 1: $r \equiv 0 \pmod{N}$ (i.e., $k=m$ mod $N$).

Then $e^{j 2\pi r n/N} = e^{0} = 1$ for all $n$. The sum equals $\sum_{n=0}^{N-1} 1 = N$.  ■

Case 2: $r \not\equiv 0 \pmod{N}$.

Let $\alpha = e^{j 2\pi r/N}$. Since $r$ is not a multiple of $N$, $\alpha \neq 1$. The sum is a finite geometric series:

Now $\alpha^N = e^{j 2\pi r N/N} = e^{j 2\pi r} = 1$ (since $r$ is an integer and Euler's formula gives $e^{j 2\pi r} = \cos(2\pi r) + j\sin(2\pi r) = 1$). Therefore the numerator $1 - \alpha^N = 1 - 1 = 0$, while the denominator $1 - \alpha \neq 0$. Hence $S = 0$.  ■

Consequence — IDFT inverts DFT. Start from the DFT:

Substitute into the IDFT candidate formula and exchange the order of summation:

The IDFT exactly recovers $x[n]$ for each $n = m$.  ■

1.4 DFT as Matrix Multiplication

Writing the DFT in full for all $k$ simultaneously reveals that it is simply a linear transformation representable by a single DFT matrix $\mathbf{W} \in \mathbb{C}^{N \times N}$.

where the vectors are $\mathbf{x} = [x[0], x[1], \ldots, x[N-1]]^T$ and $\mathbf{X} = [X[0], X[1], \ldots, X[N-1]]^T$, and the $(k, n)$ entry of $\mathbf{W}$ (using 0-based indexing) is:

For $N = 4$, the DFT matrix expands explicitly as (using $j = \sqrt{-1}$):

Note $W_4^0 = 1$, $W_4^1 = e^{-j\pi/2} = -j$, $W_4^2 = e^{-j\pi} = -1$, $W_4^3 = e^{-j3\pi/2} = j$. Each row $k$ of $\mathbf{W}_N$ is a complex sinusoid at frequency $k/N$ cycles/sample. The DFT is the inner product of the input vector with each row — i.e., the projection of $x[n]$ onto each basis sinusoid.

Inverse DFT matrix:

where $\mathbf{W}^H$ denotes the conjugate transpose. This follows directly from the orthogonality theorem: $\mathbf{W}^H \mathbf{W} = N \mathbf{I}$, so $\mathbf{W}^{-1} = \frac{1}{N}\mathbf{W}^H$.

Computational cost of direct matrix multiplication: Each of the $N$ output bins requires $N$ complex multiplications and $N-1$ complex additions. Total: $O(N^2)$ complex multiplications. For $N = 1024$: roughly $10^6$ multiplications. For $N = 4096$: $\sim 1.7 \times 10^7$. This is why direct computation becomes prohibitive for large $N$, motivating the FFT.

1.5 The FFT Algorithm — Cooley-Tukey Radix-2 DIT

The Fast Fourier Transform (FFT) is not a different mathematical transform — it computes the exact same DFT, but exploits the special structure of the DFT matrix $\mathbf{W}$ to reduce the arithmetic to $O(N \log_2 N)$ operations. The most common variant is the Cooley-Tukey radix-2 decimation-in-time (DIT) algorithm, published in 1965 (though the core idea was known to Gauss in 1805).

1.5.1 Divide and Conquer — The Danielson-Lanczos Lemma

Assume $N = 2^m$ (a power of 2). Split the input sequence into two $N/2$-point sub-sequences — the even-indexed samples and the odd-indexed samples:

The DFT can be rewritten as:

Using $W_N^{2rk} = e^{-j 2\pi (2r)k/N} = e^{-j 2\pi rk/(N/2)} = W_{N/2}^{rk}$:

where $E[k]$ and $O[k]$ are the $N/2$-point DFTs of the even and odd sub-sequences, respectively. The factor $W_N^k$ is called the twiddle factor.

1.5.2 The Butterfly and Periodicity

Since $E[k]$ and $O[k]$ are $N/2$-periodic, we have:

and $W_N^{k+N/2} = W_N^k \cdot W_N^{N/2} = -W_N^k$. Therefore:

These two equations computed together for each $k = 0, \ldots, N/2 - 1$ form a butterfly operation: two inputs $(E[k], O[k])$, one complex multiplication by $W_N^k$, one addition, one subtraction — yielding two outputs.

1.5.3 Bit-Reversal Permutation

The DIT FFT requires the input to be in bit-reversed order. For $N=8$, the natural indices $0{-}7$ in binary are $000, 001, 010, 011, 100, 101, 110, 111$. Reversing the 3-bit binary representations gives: $000, 100, 010, 110, 001, 101, 011, 111$ → indices $0, 4, 2, 6, 1, 5, 3, 7$. The butterfly structure then processes interleaved sub-sequences at each stage.

1.5.4 Complexity Analysis

The speedup grows without bound as $N$ increases. For 5G NR OFDM with $N = 4096$ (FR1 maximum), the FFT is $\sim 683\times$ faster than direct computation — making real-time OFDM processing practical on embedded hardware.

1.5.5 Why $N = 2^m$ is Preferred

The radix-2 Cooley-Tukey algorithm applies only when $N$ is a power of 2, enabling $\log_2 N$ recursion levels each with $N/2$ butterflies. For other $N$:

• Radix-4, radix-8 FFT: $N = 4^m$ or $8^m$ — even fewer multiplications due to larger butterflies (radix-4 saves $\sim 25\%$ mults vs radix-2).
• Mixed-radix FFT: $N = p \cdot q$ (composite) — split into $p$-point and $q$-point sub-DFTs. FFTW uses this for arbitrary $N$.
• Prime-length FFT: Rader's algorithm or Bluestein's chirp-z transform for prime $N$. Much less efficient.
• 5G NR practice: FFT sizes are always powers of 2 (128 to 4096), chosen to match the subcarrier-count requirement while enabling efficient radix-2/4/8 FFT implementations in hardware (FPGAs, ASICs).

1.6 IFFT — The Inverse FFT and Its Role in OFDM

The IDFT is:

Observe that this is the same computational structure as the DFT, but with the twiddle factors conjugated ($e^{+j\ldots}$ vs $e^{-j\ldots}$) and with a $1/N$ scaling. The IFFT is therefore computed by:

1. Conjugate the input: $X^*[k]$
2. Apply the standard FFT to get $Y[n] = \text{FFT}(X^*[k])$
3. Conjugate the output and scale: $x[n] = Y^*[n] / N$

Alternatively, IFFT = complex-conjugate the twiddle factors in the FFT butterfly and divide by $N$ at the end. Most hardware and software implement the FFT and IFFT with a shared butterfly core, differing only in the sign of the twiddle factor angle and the scaling.

1.7 Key Properties of the DFT / FFT

All properties below are stated for the $N$-point DFT and use the notation $x[n] \xleftrightarrow{\text{DFT}} X[k]$. Index arithmetic is modulo $N$.

Property 1 — Linearity

The DFT is a linear operator. This follows immediately from the matrix form $\mathbf{W}(a\mathbf{x}_1 + b\mathbf{x}_2) = a\mathbf{W}\mathbf{x}_1 + b\mathbf{W}\mathbf{x}_2$. Consequence: OFDM subcarriers are modulated and demodulated independently using superposition.

Property 2 — Time-Shift $\Rightarrow$ Phase Rotation in Frequency

A circular shift of $n_0$ samples in time multiplies every DFT bin by a linear phase ramp $e^{-j 2\pi k n_0/N}$. The magnitude spectrum $|X[k]|$ is unchanged — only the phase rotates. In OFDM, the cyclic prefix exploits this property: a channel delay $\tau$ introduces a phase shift $e^{-j 2\pi k \tau/N}$ per subcarrier, which is corrected by a per-subcarrier complex division (one-tap equalizer) after the FFT at the receiver.

Property 3 — Frequency-Shift $\Rightarrow$ Phase Rotation in Time (Modulation)

Multiplying by a complex exponential in time circularly shifts the spectrum. This is the discrete-time analog of the AM modulation theorem and is used in frequency-domain equalization and carrier offset estimation.

Property 4 — Circular Convolution Theorem (most important for OFDM)

where $\circledast$ denotes circular (cyclic) convolution of length $N$:

Key consequence for OFDM: A multipath channel of length $L$ (impulse response $h[n]$) performs linear convolution on the transmitted signal. By appending a cyclic prefix of length $\geq L-1$ to each OFDM symbol, the linear convolution becomes effectively circular over the FFT window. The FFT at the receiver then converts this circular convolution to a pointwise multiplication: $Y[k] = H[k] \cdot X[k]$. Channel equalization reduces to $N$ independent scalar divisions — one per subcarrier — instead of matrix inversion. This is the core computational advantage of OFDM in frequency-selective channels.

Property 5 — Parseval's Theorem (Energy Conservation)

The total energy of the time-domain signal equals the total energy of the frequency-domain signal divided by $N$ (due to the $1/N$ convention). In the unitary DFT convention, the $1/N$ factor is absent and energy is exactly preserved: $\|\tilde{\mathbf{W}}\mathbf{x}\|^2 = \|\mathbf{x}\|^2$. Parseval's theorem is used in OFDM PAPR (Peak-to-Average Power Ratio) analysis and signal-to-noise ratio calculations.

Property 6 — Conjugate Symmetry (Real Inputs)

If $x[n]$ is real-valued, then:

The DFT is Hermitian symmetric: bins $k$ and $N-k$ are complex conjugates. Only the first $N/2 + 1$ bins are independent. This is exploited in the real FFT (rfft) which computes only bins $0$ through $N/2$ in roughly half the operations of a complex FFT. In 5G NR downlink, the OFDM waveform is real-valued after RF upconversion (mixing with carrier), though the baseband model uses complex I+Q representation where this symmetry does not apply.

Property 7 — Circular Shift in Frequency → Multiplication in Time

1.8 OFDM Connection — How IFFT Creates and FFT Demodulates Subcarriers

OFDM (Orthogonal Frequency-Division Multiplexing) is a direct engineering application of the IDFT/DFT pair. The insight is elegant: the IDFT formula for OFDM modulation and the DFT formula for demodulation are computationally identical to the IFFT/FFT algorithms already described.

1.8.1 OFDM Transmitter (IFFT-based)

Given $N$ complex data symbols $\{X[k]\}_{k=0}^{N-1}$ (one per subcarrier, e.g., QAM points), the continuous-time OFDM signal over one symbol period $T = N \cdot T_s$ would ideally be:

where $\Delta f = 1/T$ is the subcarrier spacing (in Hz). At the $n$-th sample time $t = nT_s = n/(N\Delta f)$:

This is exactly the IDFT (up to the $1/N$ normalization). The IFFT simultaneously generates all $N$ subcarrier sinusoids and sums them — creating the composite time-domain waveform in a single $O(N \log N)$ operation.

1.8.2 Cyclic Prefix Insertion

After the IFFT, the last $N_{CP}$ samples of $s[n]$ are prepended to the symbol:

followed by the original $N$ samples. Total transmitted length: $N + N_{CP}$. This converts the linear channel convolution into circular convolution (as explained in §1.7, Property 4), enabling simple one-tap frequency-domain equalization.

1.8.3 OFDM Receiver (FFT-based)

After removing the cyclic prefix, the received signal (assuming AWGN channel with frequency-selective fading $H[k]$) in the frequency domain is:

where $H[k] = \text{FFT}\{h[n]\}$ is the channel frequency response at subcarrier $k$ and $Z[k]$ is the additive noise. The FFT at the receiver demodulates all $N$ subcarriers simultaneously in $O(N \log N)$ operations.

Equalization then recovers the data symbol on each subcarrier independently:

1.9 Numerical Example — $N = 8$, Rectangular Pulse

Let $x[n] = 1$ for $n = 0, 1, 2, 3$ and $x[n] = 0$ for $n = 4, 5, 6, 7$ (a length-4 rectangular window in an $N = 8$ DFT). Compute $X[k]$ analytically.

This is a geometric series with ratio $\alpha = e^{-j\pi k/4}$:

using the identity $\sum_{n=0}^{M-1} \alpha^n = e^{j(M-1)\phi/2} \cdot \sin(M\phi/2)/\sin(\phi/2)$ where $\phi = -\pi k/4$.

Observations: (1) $X[0] = 4$ is the sum of all samples (DC component). (2) $X[4] = 0$ exactly — the Nyquist bin is zero because the rectangular pulse has a null at $f = f_s/2$. (3) The spectrum is Hermitian: $X[k] = X^*[N-k]$ since $x[n]$ is real. (4) The $|X[k]|$ envelope follows a Dirichlet kernel (discrete sinc) shape.

1.10 Interactive DFT Magnitude Spectrum

The chart below shows the DFT of $x[n] = \cos\!\left(\frac{2\pi \cdot 3 \cdot n}{N}\right) + 0.5\cos\!\left(\frac{2\pi \cdot 1 \cdot n}{N}\right)$ for $N = 16$ samples. Analytically, we expect spectral peaks at bins $k = 1, 3$ (positive frequencies) and $k = 13, 15$ (corresponding negative frequencies), with magnitudes $N/2 = 8$ for the $k=3$ tone and $N/4 = 4$ for the $k=1$ tone.

1.11 Spectral Leakage, the Dirichlet Kernel, and Windowing

The DFT implicitly assumes the $N$-sample block is one period of a periodic signal. When the signal is not an integer number of cycles within the DFT window (i.e., the frequency does not fall exactly on a bin), energy "leaks" into adjacent bins — a phenomenon called spectral leakage.

The mathematical origin: the DFT of a finite-length signal is equivalent to the DTFT of the signal multiplied by a rectangular window $w[n] = 1$ for $0 \leq n \leq N-1$. In frequency, this multiplication becomes circular convolution of the ideal spectrum with the DFT of the rectangle — the Dirichlet kernel:

$|D_N|$ has a main lobe of width $4\pi/N$ and sidelobes at approximately $-13\,\text{dB}$ below the main lobe peak. Spectral leakage is the convolution of any sharp spectral feature with these sidelobes.

Windowing applies a time-domain taper $w[n]$ to the signal before the DFT, trading main-lobe width (frequency resolution) for reduced sidelobe level (reduced leakage):

1.12 Zero-Padding and DFT Interpolation

Appending $M$ zeros to $x[n]$ before computing an $(N+M)$-point DFT does not add new information but produces a denser sampling of the DTFT — effectively interpolating the frequency spectrum:

The frequency resolution (bin spacing) becomes $f_s/(N+M)$ instead of $f_s/N$. Zero-padding is used to: (1) make $N$ a power of 2 for the FFT, (2) improve visual interpolation of spectra, (3) compute circular cross-correlation via the overlap-add method for linear convolution.

2.1  CP-OFDM Baseband Signal Model

The continuous-time CP-OFDM baseband signal for a single OFDM symbol is the superposition of N complex exponentials, one per subcarrier:

2.2  Multicarrier Structure — N Parallel Narrow-Band Channels

OFDM is a multicarrier modulation scheme. Rather than transmitting one symbol per use of the channel at the full bandwidth B (as in single-carrier BPSK/QAM), OFDM divides B into N narrow sub-bands, each of width ≈ Δf, and transmits one QAM symbol per sub-band simultaneously:

The serial-to-parallel (S/P) converter at the transmitter maps a stream of QAM symbols onto the N frequency-domain inputs of the IFFT. After the IFFT and DAC, all N tones are present in the transmitted signal simultaneously — the channel sees a wideband signal, but the DSP treats it as N independent narrowband channels.

2.3  Orthogonality Proof

The fundamental reason CP-OFDM works without inter-subcarrier interference (ICI) is that the set of complex exponentials \(\{\phi_k(t) = e^{j2\pi k \Delta f t}\}\) forms an orthogonal basis over the interval \([0, T_u]\).

Case 1: m = n

Case 2: m ≠ n  (let \(p = m - n \in \mathbb{Z} \setminus \{0\}\))

Substituting \(\Delta f \cdot T_u = 1\):

Combining both cases with the Kronecker delta \(\delta_{mn}\):

2.4  Subcarrier Spacing & Symbol Duration — 5G NR Numerologies

The fundamental constraint is the time-frequency uncertainty relation for OFDM:

5G NR defines a set of numerologies indexed by \(\mu \in \{0,1,2,3,4\}\), where \(\Delta f = 2^\mu \times 15\,\text{kHz}\). Higher \(\mu\) gives larger spacing (shorter symbols), trading time resolution for Doppler robustness.

2.5  OFDM Symbol with Cyclic Prefix (CP)

After the IFFT produces N time-domain samples \(\{s_0, s_1, \ldots, s_{N-1}\}\), a cyclic prefix of length \(N_{CP}\) samples is prepended:

The total symbol duration is:

where \(f_s = N \cdot \Delta f\) is the sampling rate.

2.6  Discrete-Time CP-OFDM — the IFFT Implementation

In practice, the continuous-time model is implemented entirely in discrete time. The N-point IFFT of the frequency-domain symbols \(\{S_0, \ldots, S_{N-1}\}\) yields the time-domain samples:

This is the N-point IDFT. The factor \(1/N\) is an implementation convention (some implementations absorb it into the FFT scaling). Comparing to the continuous-time model with \(t = n T_s / N = n / f_s\):

The discrete and continuous models are identical at the sample points.

The complexity advantage over a bank of N matched filters is enormous: \(\mathcal{O}(N \log_2 N)\) for the FFT versus \(\mathcal{O}(N^2)\) for direct DFT computation. For \(N = 4096\) (5G NR, 100 MHz FR1), this is a factor of \(\sim\!341\times\) reduction in multiply-accumulate operations.

2.7  CP-OFDM Spectrum & Spectral Efficiency

Each subcarrier k, viewed in isolation, is a finite-duration complex tone of duration \(T_u\). Its Fourier transform is a sinc function:

where \(\operatorname{sinc}(x) = \sin(\pi x)/(\pi x)\). The main lobe of subcarrier k is centred at \(f = k \Delta f\) with first nulls at \(f = k\Delta f \pm \Delta f\) — precisely at the centres of adjacent subcarriers. This is the geometric reason for orthogonality: each subcarrier's sinc passes through zero at all other subcarrier frequencies.

Spectral Efficiency

The CP wastes \(N_{CP}/(N + N_{CP})\) of the time resource. The spectral efficiency (fraction of time used for useful data) is:

2.8  Resource Grid — Time-Frequency Structure

The transmitted signal is organized into a resource grid in the time-frequency plane:

2.9  CP-OFDM vs Plain OFDM — Why CP is Essential

Without the CP, the OFDM signal would suffer from two simultaneous impairments in a multipath channel:

3.1 Multipath Propagation Model

In a wireless channel the transmitted signal arrives at the receiver via multiple paths — reflections, diffractions and scattering off buildings, terrain and vehicles. Each path adds a scaled, delayed copy of the signal. The baseband-equivalent received signal is modelled as a tapped-delay line:

where \(h_l \in \mathbb{C}\) is the complex gain of the \(l\)-th path, \(\tau_l\) is its propagation delay, \(L\) is the total number of resolvable paths, and \(n(t)\) is additive white Gaussian noise (AWGN). In the continuous-time domain the channel is fully described by its channel impulse response (CIR):

The key channel parameter for OFDM design is the maximum delay spread \(\tau_{\max} = \max_l \tau_l - \min_l \tau_l\). From it we derive the coherence bandwidth — the frequency range over which the channel response is approximately flat:

3.2 Inter-Symbol Interference (ISI) — Without CP

Consider two consecutive OFDM symbols, each of duration \(T_u = N/f_s\) seconds (where \(N\) is the FFT size and \(f_s\) is the sample rate). Without a guard interval, the \(m\)-th received sample of symbol \(i\) is:

If the channel length \(L > 1\) (i.e. \(\tau_{\max} > 0\)), delayed copies of the previous symbol \(s_{i-1}\) spill into the current symbol's observation window. This Inter-Symbol Interference destroys the orthogonality of the DFT basis and cannot be removed with a simple single-tap equalizer.

3.3 Inter-Carrier Interference (ICI) — Linear vs. Circular Convolution

Even if ISI is ignored, linear convolution in the time domain breaks subcarrier orthogonality. The DFT of a linear convolution is not the pointwise product of the individual DFTs:

As a result, subcarrier \(k\) receives energy from all other subcarriers \(k' \neq k\) — this is ICI. Each received DFT bin becomes:

where \(I_{kk'}\) are inter-carrier leakage coefficients whose magnitude depends on how far the received window is misaligned with the circular structure of the channel.

3.4 CP Insertion and Removal — Proof of Circularity

The cyclic prefix is formed by copying the last \(N_{CP}\) samples of the IFFT output and prepending them. The transmitted block of length \(N + N_{CP}\) is:

Transmitter:
  s[0..N-1]  ← IFFT(S[0..N-1])           // N-point IFFT
  s_tx[0..N_CP-1]   ← s[N-N_CP .. N-1]   // copy last N_CP samples
  s_tx[N_CP..N+N_CP-1] ← s[0..N-1]       // append payload
  transmit s_tx  (length N + N_CP)

Receiver:
  r_rx[0..N+N_CP-1] ← received block
  discard r_rx[0..N_CP-1]                 // remove CP
  r[0..N-1] ← r_rx[N_CP..N+N_CP-1]       // N samples for DFT
  R[0..N-1] ← FFT(r[0..N-1])

Proof: CP Enforces Circular Convolution

Assume the channel has \(L\) taps: \(h[0], h[1], \ldots, h[L-1]\) and \(N_{CP} \geq L - 1\). After the CP is stripped, the \(n\)-th received sample is:

For \(n - l \geq 0\): \(\tilde{s}[n + N_{CP} - l] = s[n - l]\). For \(n - l < 0\): the sample falls in the CP region, but because \(N_{CP} \geq L - 1\), it equals \(s[N + n - l]\) — exactly what circular indexing \(s[(n-l) \bmod N]\) gives. Therefore:

Taking the N-point DFT of both sides, and using the circular-convolution theorem:

The channel is perfectly diagonalised: each subcarrier \(k\) sees only a single complex multiplication by \(H[k]\) — no ISI, no ICI.

3.5 Single-Tap Frequency-Domain Equalization

Because \(R[k] = H[k]\,S[k] + W[k]\), recovering \(S[k]\) requires only a scalar division (or multiplication by a weight) per subcarrier. Two common equalizer choices:

ZF perfectly cancels the channel but amplifies noise at deep fades (\(|H[k]| \approx 0\)). MMSE adds a noise-regularisation term \(\sigma_w^2/\sigma_s^2\) in the denominator, trading a small residual bias for reduced noise enhancement. MMSE reduces to ZF at high SNR.

3.6 CP Overhead and 5G NR Numerology

The CP occupies \(N_{CP}\) out of every \(N + N_{CP}\) transmitted samples. The CP overhead fraction is:

In 5G NR the FFT size and CP length scale with the subcarrier spacing \(\Delta f = 2^\mu \times 15\text{ kHz}\) (numerology index \(\mu\)). Normal CP values for the reference sample rate of 30.72 MHz (\(N = 2048\) at 15 kHz):

3.7 CP Length Design: Matching Delay Spread to Numerology

The fundamental constraint is \(T_{CP} \geq \tau_{\max}\). Choosing a larger \(\Delta f\) (higher numerology) reduces \(T_{CP}\) and requires a lower \(\tau_{\max}\) — fine for indoor / mmWave deployments but problematic for rural macro cells.

3.8 OFDM Channel Model: Per-Subcarrier Fading

With CP in place the channel is fully described by \(H[k]\) — the channel frequency response sampled at each subcarrier. In a Rayleigh fading environment, each \(H[k] \sim \mathcal{CN}(0,1)\) independently when subcarrier spacing \(\Delta f \gg B_c^{-1}\). In practice, adjacent subcarriers are correlated over a coherence bandwidth of roughly \(B_c \approx 1/(5\tau_{\max})\):

where \(R_H(f)\) is the channel's frequency correlation function and \(\tau_{\text{rms}}\) is the RMS delay spread. This frequency correlation is exploited by:

• Channel estimation interpolation — pilots on every \(P\)-th subcarrier, interpolate over \(P\) subcarriers if \(P \cdot \Delta f \ll B_c\).
• Frequency-domain MIMO precoding — beamforming weights vary slowly across subcarriers.
• Link adaptation — CQI feedback per PRB (12 subcarriers) assumes flat fading within a PRB when \(12\Delta f \ll B_c\).

3.9 Guard Interval Alternatives: ZP-OFDM and Others

The CP is not the only guard interval strategy. Zero-Padding OFDM (ZP-OFDM) appends \(N_{ZP}\) zeros after the IFFT block instead of a cyclic prefix:

4.1 PAPR Definition

The Peak-to-Average Power Ratio quantifies how much the instantaneous power of a transmitted signal can exceed its average power. For a continuous-time OFDM signal s(t):

An OFDM symbol with N subcarriers, each bearing unit-power:

Worst case: all N subcarriers align in phase at the same instant, so the instantaneous amplitude is N times the per-subcarrier amplitude:

4.2 Statistical Analysis of PAPR

For large N, the Central Limit Theorem (CLT) applies: the sum of many independent random variables approaches a Gaussian distribution. The real and imaginary parts of s(t) each become approximately i.i.d. Gaussian, so the envelope |s(t)| follows a Rayleigh distribution.

The instantaneous power |s(t)|² follows an exponential distribution with mean P_avg. Treating the N sub-samples as independent (Nyquist rate), the Complementary CDF (CCDF) of PAPR is:

Operating PAPR at 0.1% outage (CCDF = 10⁻³): solving 1 − (1 − e^−γ)^N = 10⁻³:

4.3 Why PAPR Matters — The HPA Problem

A High Power Amplifier (HPA) is characterised by a linear region up to its saturation point P_sat, beyond which the output clips and severe nonlinear distortion is introduced. The transmit chain must operate with sufficient back-off to keep the signal within the linear region.

where OBO is the Output Back-Off. For reliable operation, OBO must equal (or exceed) the PAPR at the required outage level.

Quantifying Efficiency Loss

For a Class A amplifier, the theoretical drain efficiency is:

At saturation (OBO = 0 dB): η_A = 50%. With a 10 dB back-off:

Class B amplifiers have a more favourable back-off characteristic:

Peak efficiency ≈ 78.5% at saturation, dropping to ≈ 24.8% at 10 dB OBO — still painful, but significantly better than Class A.

4.4 PAPR Reduction Techniques

Six major categories of PAPR reduction exist, each with distinct trade-offs in complexity, spectral efficiency, and performance.

4.5 DFT-s-OFDM (SC-FDMA) — PAPR Advantage in Detail

Transmitter Architecture

The DFT-s-OFDM transmitter adds a single DFT precoding stage before the conventional OFDM IFFT:

The cascade of M-point DFT followed by the N-point IFFT is equivalent to transmitting M data symbols as a single-carrier signal occupying a bandwidth of M · Δf. Mathematically, the time-domain output is:

Substituting, s[n] reduces to a linearly-modulated single-carrier signal (with rectangular pulse-shaping), explaining the dramatically lower PAPR.

PAPR Numbers

Why 5G NR Uses Both Waveforms

4.6 Waveform Comparison: CP-OFDM vs DFT-s-OFDM

Long Term Evolution (LTE), standardised in 3GPP Release 8 (2008) and refined through Releases 9–15, brought OFDM from Wi-Fi into wide-area cellular networks for the first time. The downlink uses OFDMA (Orthogonal Frequency Division Multiple Access) while the uplink uses SC-FDMA (Single-Carrier FDMA, also called DFT-s-OFDM) — a deliberate asymmetry driven by the need to conserve UE battery life. Understanding LTE's parameter choices is the essential prerequisite for 5G NR's flexible numerology (§8).

7.1 LTE OFDM Parameters

LTE fixes the subcarrier spacing at 15 kHz for all channel bandwidths. This was chosen to balance Doppler tolerance (wider spacing is better) against multipath tolerance (wider spacing shortens the useful symbol time, reducing CP efficiency). 15 kHz maps to a useful symbol duration of exactly \(T_u = 1/\Delta f = 66.7\;\mu\text{s}\).

7.2 LTE OFDMA — Downlink Architecture

In the LTE downlink, the eNodeB (base station) generates all subcarriers simultaneously via a single N-point IFFT. Multiple UEs are multiplexed by assigning each a non-overlapping set of Resource Blocks (RBs) within each 1 ms subframe — this is Orthogonal Frequency Division Multiple Access.

Resource Block Definition

Frequency-Selective Scheduling

At each TTI (1 ms), the eNodeB scheduler assigns RBs to UEs based on Channel Quality Indicator (CQI) feedback. A UE reports the CQI it measures on each subband (a group of RBs). The scheduler exploits multi-user diversity: different UEs experience fading peaks at different frequencies, so the aggregate system capacity is maximised by assigning each subband to whichever UE sees it best — the proportional-fair or maximum C/I criterion.

7.3 SC-FDMA (DFT-s-OFDM) — Uplink Architecture

Why Not OFDMA in the Uplink?

OFDMA's chief drawback is a high Peak-to-Average Power Ratio (PAPR) — typically 8–12 dB (see §4). A high PAPR forces the UE power amplifier to operate far below its saturation point (large backoff), drastically reducing power efficiency. For a battery-powered UE, this directly reduces talk time. The eNodeB is a mains-powered infrastructure node where PAPR is a manageable engineering problem; the UE is not.

DFT-s-OFDM Signal Processing Chain

Input bits
  → QAM mapper (QPSK / 16-QAM / 64-QAM)  →  M data symbols: d[0..M-1]
  → M-point DFT  →  D[k] = Σ_{m=0}^{M-1} d[m] · e^{-j2πmk/M},  k=0..M-1
  → Subcarrier mapping (M occupied out of N total, N > M)
        LFDMA: map D[k] to contiguous subcarriers [k_0 .. k_0+M-1]
        IFDMA: map D[k] to every (N/M)-th subcarrier (not used in LTE)
  → N-point IFFT  →  time-domain single-carrier-like waveform  s[n]
  → Insert cyclic prefix (same structure as OFDM)
  → DAC + RF → air interface
    

The key insight: the DFT "pre-spreads" the M QAM symbols across M subcarriers so that the final IFFT output looks like a single-carrier signal. The time-domain signal has the envelope statistics of single-carrier (low PAPR) rather than a sum of many independent sinusoids (high PAPR).

M Must Be "Highly Composite"

The M-point DFT is implemented as an FFT. For computational efficiency, M must factor into small primes. LTE mandates that M (the number of assigned subcarriers, always a multiple of 12) must be expressible as \(M = 2^a \cdot 3^b \cdot 5^c\) with \(a,b,c \geq 0\). This is called a "good" or highly-composite number constraint. Valid values in LTE include 12, 24, 36, 48, 60, 72, 96, 120, 144, 180, … up to 1200. Any M that requires a prime factor > 5 is forbidden, which means RB allocations of 7, 11, 13 RBs (= 84, 132, 156 subcarriers) are not schedulable.

7.4 LTE Reference Signals

7.5 LTE MIMO and OFDM Interaction

OFDM converts a wideband frequency-selective MIMO channel into a bank of narrowband flat-fading MIMO subchannels — one per subcarrier. This is the most elegant property of OFDM for MIMO: the spatial processing (precoding, detection) can be performed independently per subcarrier.

Space-Frequency Block Coding (SFBC) in LTE TM2 encodes pairs of symbols across two antenna ports and two adjacent subcarriers, achieving 2nd-order diversity without CSI at the transmitter. It is the LTE equivalent of Alamouti coding but mapped to the frequency dimension (rather than time) to avoid the time variation between symbols.

7.6 LTE Capacity — Shannon Analysis

For LTE 20 MHz with 2×2 MIMO (2 independent spatial streams):

Practical LTE Peak vs Theoretical

7.7 LTE vs 5G NR — Comparison Preview

Full 5G NR treatment follows in §8. This teaser table highlights the key differences.

7.8 Interactive Charts

9.1 CP-OFDM Limitations That Motivate Alternatives

CP-OFDM has dominated 4G and 5G NR because of its elegant frequency-domain equalization (one complex multiply per subcarrier), but four structural weaknesses motivate the search for alternatives.

9.1.1 Cyclic-Prefix Overhead

The CP of length \(N_{CP}\) is a copy of the last \(N_{CP}\) samples of each OFDM symbol of length \(N\). The spectral efficiency penalty is:

In 5G NR numerology \(\mu=0\) (SCS = 15 kHz) the normal CP is 144 samples over a 2048-sample FFT, giving \(\eta = 2048/(2048+144) \approx 93.4\%\). At \(\mu=4\) (SCS = 240 kHz) the extended CP reduces efficiency further. For ultra-dense small-cell deployments, even this 6–7% loss translates to meaningful capacity reduction.

9.1.2 High Out-of-Band Emissions (Sinc Sidelobe Problem)

The rectangular window applied to each OFDM symbol has a frequency-domain response of the form \(\mathrm{sinc}(fT)\). The first sidelobe is only \(-13.3\) dB below the main lobe. For a multi-carrier signal the composite power spectral density decays as:

This slow \(\sim 1/f^2\) rolloff forces regulators to mandate wide guard bands around the occupied spectrum — 0.5–1 MHz in LTE/5G NR — wasting radio resources. Adjacent-channel interference into other operators or asynchronous users (unlicensed, NB-IoT) is a direct consequence.

9.1.3 Peak-to-Average Power Ratio (PAPR)

An OFDM signal with \(N\) independently modulated subcarriers can constructively superpose. In the worst case:

For \(N=1024\) this is 30 dB — far in excess of what any practical power amplifier can tolerate linearly. Statistical measures (complementary CDF at \(10^{-3}\)) give PAPR \(\approx 10\text{–}11\) dB for QPSK/16-QAM, still requiring significant power back-off that degrades energy efficiency.

9.1.4 Sensitivity to Doubly-Dispersive Channels

CP-OFDM restores orthogonality only when the channel delay spread \(\tau_{\max} \le T_{CP}\) and the Doppler spread \(\nu_{\max} \ll \Delta f\). In vehicular or LEO-satellite channels both conditions can be violated simultaneously. Inter-carrier interference (ICI) from Doppler shifts destroys the diagonal structure of the channel matrix, requiring costly equalizers and reducing throughput.

9.2 FBMC — Filter Bank Multi-Carrier

FBMC replaces the rectangular pulse of OFDM with a well-designed prototype filter \(p(t)\) applied independently to each subcarrier. The transmitted signal is:

where \(d_{k,m}\) are real-valued OQAM symbols, \(T\) is the symbol period, \(\Delta f = 1/T\) is subcarrier spacing, and \(\varphi_{k,m} = \tfrac{\pi}{2}(k+m)\) is the phase offset used in Offset-QAM.

9.2.1 Prototype Filter Design — PHYDYAS Filter

The most widely studied prototype filter for FBMC is the PHYDYAS filter, specified by its frequency-domain coefficients. For overlapping factor \(K=4\) (filter spans 4 multicarrier symbol periods) the frequency samples are:

The prototype filter impulse response in continuous time has length \(L_p = K\cdot N\) samples (where \(N\) is the FFT size). The resulting power spectral density sidelobe is suppressed to below \(-40\) dB, compared to only \(-13\) dB for the rectangular window (OFDM).

9.2.2 OQAM — Offset QAM

The fundamental constraint of FBMC is that perfect reconstruction without a CP requires the transmit basis functions to satisfy the Nyquist condition only in the real field. Complex QAM symbols produce unavoidable intrinsic imaginary interference from neighboring subcarriers. OQAM circumvents this by transmitting the in-phase (I) and quadrature (Q) components of each QAM symbol on staggered time grids separated by \(T/2\):

where \(c_{k,n}\) is the complex QAM symbol on subcarrier \(k\) at time index \(n\). The real orthogonality condition for FBMC-OQAM is:

The imaginary interference \(\mathrm{Im}\{\langle g_{k,m}, g_{k',m'}\rangle\}\) from the nearest time-frequency neighbors is non-zero but known — it can be cancelled with auxiliary pilots or by treating it as structured noise in MMSE equalization.

9.3 f-OFDM — Filtered OFDM

Filtered OFDM (f-OFDM) is a pragmatic variant that applies a single time-domain filter to each sub-band of contiguous subcarriers, rather than filtering at the individual-subcarrier level. The baseband model for sub-band \(b\) is:

where \(x_b(t)\) is the conventional CP-OFDM waveform for sub-band \(b\) and \(h_b(t)\) is a raised-cosine or windowed-sinc sub-band filter with a rolloff factor \(\beta\) and passband equal to the sub-band bandwidth.

9.3.1 Mixed Numerology in 5G NR

A key motivation for f-OFDM is the ability to multiplex different subcarrier spacings (\(\mu = 0, 1, 2, \ldots\) corresponding to SCS 15, 30, 60 kHz, …) in the same wideband carrier without mutual interference. Each numerology occupies its own sub-band and is filtered independently:

At the filter transition band (a few subcarriers wide), inter-sub-band interference (ISBI) exists. In 5G NR this is managed by the guard sub-band mechanism — typically 1–2 subcarriers are left unused at sub-band edges.

9.3.2 f-OFDM vs UFMC

9.4 UFMC — Universal Filtered Multi-Carrier

UFMC filters each resource block (RB) or small group of RBs individually using a Dolph-Chebyshev filter. The transmitted signal is the sum of per-RB filtered OFDM bursts:

where \(\mathbf{d}_i \in \mathbb{C}^{N_i}\) is the QAM symbol vector for RB \(i\), \(\mathbf{V}_i\) is the \(N \times N_i\) IDFT submatrix for the \(N_i\) active subcarriers of RB \(i\), and \(\mathbf{F}_i\) is the convolution (Toeplitz) matrix of the Dolph-Chebyshev filter of length \(L\). The total transmitted block length is \(N + L - 1\) samples — no CP is inserted.

9.4.1 Dolph-Chebyshev Filter Properties

The Dolph-Chebyshev filter minimizes the main lobe width for a given maximum sidelobe level (or vice versa). For a prescribed sidelobe attenuation \(A\) dB, the filter order \(L\) and window coefficients are determined analytically via:

where \(T_n(\cdot)\) is the Chebyshev polynomial of the first kind and \(x_0 = \cosh\!\bigl(\tfrac{1}{L-1}\cosh^{-1}(10^{A/20})\bigr)\). Typical UFMC parameters: \(L = 73\) (for a 5G NR 12-subcarrier RB at SCS 15 kHz), sidelobe attenuation 40 dB.

9.4.2 UFMC Target Scenarios

UFMC was proposed in 5GNOW (EU FP7 project) primarily for:

• NB-IoT and M-MTC: Low-cost devices with timing uncertainty — the Chebyshev filter tolerates asynchronous users within the same sub-band better than CP-OFDM.
• Short-burst communications: Removing the CP saves \(\sim 7\%\) overhead for short packets where control overhead already dominates.
• D2D and V2X: Asynchronous peer-to-peer links where strict uplink timing synchronization is impractical.

9.5 OTFS — Orthogonal Time-Frequency Space

OTFS is a modulation scheme proposed by Hadani et al. (2017) that places information symbols in the delay-Doppler (DD) domain rather than the time-frequency (TF) plane. The fundamental insight is that for typical wireless channels, the DD domain representation is sparse: each physical scatterer corresponds to a single (delay, Doppler) tap, making the channel extremely compact regardless of mobility.

9.5.1 Delay-Doppler Signal Model

Consider an \(M \times N\) OTFS frame. Symbols \(x[l,k]\) are placed on a DD grid with delay resolution \(\Delta\tau = 1/(M\Delta f)\) and Doppler resolution \(\Delta\nu = 1/(NT)\). The modulation chain is:

\(X[n,m]\) is the time-frequency domain representation, which is then converted to a time-domain waveform via the Heisenberg transform (a standard OFDM modulator):

where \(g_{tx}(t)\) is the transmit pulse (rectangular for OTFS-CP). At the receiver, the Wigner transform demodulates back to TF, and a 2D-SFFT transforms to DD:

This is a 2D circular convolution with the DD domain channel \(h_w[l',k']\) — a sparse matrix with at most \(P\) non-zero entries (one per scattering path). This sparsity underpins OTFS's efficiency advantage for channel estimation and equalization.

9.5.2 Full Diversity Property

Every OTFS symbol \(x[l,k]\) spreads its energy across the entire time-frequency plane via the 2D-ISFFT. As a result, each symbol experiences the full diversity of the channel — all \(P\) delay-Doppler paths contribute to each received DD-domain sample. The diversity order is:

In contrast, a single CP-OFDM subcarrier experiencing fading has diversity order 1 (per subcarrier) without coding or interleaving. This is why OTFS shows dramatically better BER at high mobility — it converts fading dips into a manageable average-SNR problem.

9.5.3 Relationship to CP-OFDM and PAPR

OTFS is mathematically equivalent to CP-OFDM preceded by a 2D-ISFFT precoding matrix. This means:

• PAPR: Identical to CP-OFDM (the precoding is unitary; it redistributes symbols across TF but does not reduce peak-to-average ratio statistically).
• Implementation: An OTFS modulator can be built on top of an existing OFDM stack with a software precoding layer.
• Guard intervals: OTFS-CP uses one CP per OTFS frame (not per symbol), recovering most of the CP overhead of symbol-by-symbol OFDM.

9.5.4 DD Channel Estimation

Since the channel is sparse in DD domain (\(P \ll MN\)), pilot-based estimation requires far fewer pilots than OFDM. A single impulse pilot at DD coordinate \((l_p, k_p)\) with guard region produces:

The guard region of size \((2l_{\max}+1)\times(2k_{\max}+1)\) around the pilot isolates the channel response from data interference. The pilot overhead scales with the channel spread, not the frame size, giving significant savings for sparse doubly-dispersive channels.

9.6 Waveform Comparison

9.7 OQAM Nyquist Condition for the Prototype Filter

The prototype filter \(p(t)\) must satisfy the real Nyquist condition to guarantee zero inter-symbol and inter-carrier interference in FBMC-OQAM. In the Zak (or Wigner-Ville) domain, this condition is:

Equivalently in the frequency domain (Balian-Low constraint):

This is the half-Nyquist (power complementary) condition. The PHYDYAS filter satisfies (9.19) up to a very good approximation for \(K \geq 4\). The ISOTROPE filter satisfies it exactly with a closed-form expression.

9.8 Visualizations

9.9 Study Questions

This section provides a rigorous, balanced assessment of CP-OFDM — the waveform that underpins 4G LTE, 5G NR, Wi-Fi 4–7, and most modern broadband standards. Every advantage carries a quantified engineering justification; every disadvantage is paired with its real-world impact and the mitigations that have been standardised or researched. The section closes with a deployment decision matrix and two interactive charts.

1   Advantages of CP-OFDM

1a   Multipath Resilience via Single-Tap Equalization

The cyclic prefix converts the linear channel convolution into a circular one. In the frequency domain this means that the received subcarrier k sees a multiplicative scalar channel coefficient rather than inter-symbol interference:

Equalising each subcarrier therefore costs exactly one complex multiplication per subcarrier per OFDM symbol, regardless of the channel delay spread. Compare this with the brute-force single-carrier (SC) alternative:

where \(M\) is the constellation order and \(J\) is the number of channel taps. For a 100-tap LTE Urban-Macro channel with 64-QAM (\(M=64\)):

• OFDM: 100 complex multiplications (one per tap-equivalent subcarrier)
• SC + Viterbi: \(64^{100} \approx 10^{180}\) states — computationally impossible.

1b   Efficient FFT-Based Implementation

The modulator/demodulator is a single IFFT/FFT pair. The Cooley–Tukey radix-2 FFT reduces the multiply count from \(N^2\) (direct DFT) to \(\frac{N}{2}\log_2 N\) complex multiplications:

For N = 2048 (LTE 20 MHz / 5G NR μ=0 at 30.72 Msps):

For N = 4096 (400 MHz NR, μ=3) the savings are even larger: 8,192 vs 16,777,216 — a 2048× reduction. This makes real-time 100+ MHz OFDM feasible on a single embedded DSP or FPGA slice.

1c   Flexible Frequency-Domain Resource Allocation (OFDMA)

Because each subcarrier is modulated independently, the scheduler can allocate different subsets of subcarriers to different users, power levels, and MCS in each OFDM symbol. This enables three powerful capabilities:

1. Water-filling capacity: allocate power inversely proportional to noise level per subcarrier. Theoretical capacity approaches the Shannon limit: \[C = \sum_{k} B_\text{sc} \log_2\!\left(1 + \frac{P_k |H[k]|^2}{\sigma^2}\right)\]
2. Frequency-selective multi-user scheduling: assign subcarrier group \(k\) to the user whose channel \(|H_u[k]|\) is largest — multiuser diversity gain scales as \(\log\log K\) for \(K\) users (Knopp–Humblet theorem).
3. Non-contiguous spectrum access: subcarriers in a fragmented licensed band or cognitive-radio secondary band can be individually toggled, something impossible with a single-carrier system without a full redesign.

1d   Easy MIMO Implementation

At each subcarrier the channel is flat-fading: the MIMO channel matrix \(\mathbf{H}[k] \in \mathbb{C}^{N_r \times N_t}\) is constant across that subcarrier's bandwidth. Spatial multiplexing therefore reduces to a simple matrix equation per subcarrier:

For massive MIMO with \(N_t = 64\) transmit antennas and a single-stream UE (\(N_r = 1\)), the per-subcarrier equalizer is 64 scalar multiplications — one weight per antenna. Contrast this with a wideband single-carrier massive MIMO equalizer which requires a full \(64 \times J\)-tap matrix filter per receive antenna.

1e   Scalable Bandwidth and Numerology

The OFDM framework parameterises cleanly: subcarrier spacing \(\Delta f = 1/T_u\), FFT size \(N\), CP length \(N_\text{CP}\), and sampling rate \(f_s = N \cdot \Delta f\). 5G NR defines five numerologies sharing the same baseband architecture:

Only the FFT size, CP tap count, and sampling clock change between numerologies. The IFFT/FFT hardware IP block is identical — a major silicon cost saving.

1f   Spectral Efficiency

Orthogonality allows subcarriers to overlap in frequency while remaining inter-symbol interference-free — eliminating the guard bands required between single-carrier channels:

With adaptive modulation and coding (AMC) per subcarrier, OFDM approaches the Shannon bound. Practical LTE/NR peak spectral efficiencies (downlink):

• LTE Cat-4 SISO: ≈5.1 bit/s/Hz at 64-QAM R=4/5
• LTE Advanced 4×4 MIMO: ≈16 bit/s/Hz (4-layer 256-QAM)
• 5G NR 8×8 DL MIMO: up to ≈30 bit/s/Hz (8-layer 256-QAM 5/6)

2   Disadvantages of CP-OFDM

2a   High Peak-to-Average Power Ratio (PAPR)

An OFDM signal is the sum of \(N\) independently modulated sinusoids. By the central limit theorem the real and imaginary parts approach Gaussian distributions as \(N \to \infty\), giving a CCDF tail that decays slowly. The theoretical maximum PAPR is:

For N = 2048 this is 33 dB, but the practical 10^-4 CCDF value is 10–12 dB. The consequences for the power amplifier (PA) are severe:

A 10 dB back-off reduces PA drain efficiency from ~50% to ~5% — a 10× increase in power consumption for the same useful RF output. This is why 3GPP chose DFT-s-OFDM (SC-FDMA) for the LTE/NR uplink: the single-carrier nature reduces PAPR by 4–6 dB, cutting UE battery drain significantly.

2b   Cyclic Prefix Overhead

The CP carries no new data — it is a copy of the tail of the OFDM symbol prepended to guard against ISI. Every CP sample is "wasted" bandwidth:

Concrete values:

The overhead cannot be reduced below the channel's maximum excess delay divided by the sampling period without incurring ISI. For LTE, the normal CP covers \(N_\text{CP} \times T_s = 144 / 30.72\text{ MHz} = 4.69\;\mu\text{s}\) — sufficient for macro-cell delay spreads up to ≈1.4 km path-length difference.

2c   Sensitivity to CFO and Phase Noise

A carrier frequency offset (CFO) \(\varepsilon\) normalised to subcarrier spacing (\(\varepsilon = \delta f / \Delta f\)) breaks orthogonality, causing inter-carrier interference (ICI). The SIR from ICI is approximately:

Example: a 750 Hz CFO on a 15 kHz subcarrier grid (\(\varepsilon = 0.05\)):

This limits the useful SNR to 20.8 dB — insufficient for 256-QAM (requires >30 dB SIR). Real oscillator specs must achieve sub-10 Hz residual CFO after AFC for 256-QAM links.

Phase noise at mmWave: LO phase noise power spectral density scales as \(f_c^2\) (for a free-running VCO). At 60 GHz vs 30 GHz the phase noise is 4× worse in power (6 dB) — and the wider 120 kHz / 240 kHz subcarrier spacing of 5G NR FR2 only partially compensates. 3GPP introduced Phase Tracking Reference Signals (PTRS) in NR FR2 specifically to track and cancel low-frequency phase noise.

2d   Out-of-Band Emissions (OOB)

The OFDM time-domain symbol has a rectangular (abrupt) window. Its frequency spectrum is a sum of sinc functions, with the first sidelobe at only −13 dB relative to the main lobe:

To protect adjacent channels, guard bands must be inserted. LTE allocates approximately 10% of the channel bandwidth as guard subcarriers (e.g., 72 unused subcarriers at each edge of the 2048-point FFT in 20 MHz mode). These are in addition to any regulatory spectral mask requirements.

2e   Poor Performance in Doubly Dispersive Channels

CP-OFDM handles delay spread (frequency selectivity) via the CP, but it handles Doppler spread (time selectivity) only if the channel is quasi-static within one OFDM symbol duration \(T_u\). The ICI caused by Doppler shift \(f_D\) is approximately:

For a 5G NR μ=0 symbol at 250 km/h and 3.5 GHz: \(f_D = 250 \times 3.5 \times 10^9 / (3 \times 10^8 \times 3.6) \approx 810\;\text{Hz}\), and \(T_u = 66.7\;\mu\text{s}\), giving \(f_D T_u \approx 0.054\) and SIR ≈ 20.4 dB — borderline for 256-QAM.

When both delay spread and Doppler are significant (doubly dispersive), the CP may be insufficient (ISI) and the symbol duration may be too long (ICI) simultaneously. Increasing \(\Delta f\) (shorter symbol) reduces ICI but also reduces the maximum tolerable delay spread.

2f   High ADC/DAC Complexity for Wide Bandwidth

OFDM exploits bandwidth by increasing the number of subcarriers, which requires proportionally higher sampling rates. The Nyquist sampling rate for a 400 MHz NR carrier (μ=3, FFT=4096) with standard oversampling is:

After 2× oversampling for interpolation filtering: 983.04 Msps. ADC power consumption scales roughly as \(f_s \times 2^\text{ENOB}\), so a 12-bit ADC at 1 Gsps consumes ≈500 mW — substantial in a battery-powered UE. Contrast with NB-IoT (180 kHz, μ=0): ADC at 240 ksps draws <1 mW.

3   Net Verdict: Why OFDM Won — and Where It Falls Short

CP-OFDM won the 4G/5G standardisation process for a combination of reasons that no single-carrier alternative could simultaneously match in the mid-2000s engineering environment:

1. Equalizer simplicity: one multiplication per subcarrier vs exponential SC complexity enabled real-time 20 MHz 2×2 MIMO on the <65 nm silicon of 2008.
2. OFDMA multi-user flexibility: sub-carrier granularity enabled the LTE scheduler architecture that drives the 3–5× throughput gains of LTE over HSPA.
3. MIMO synergy: per-subcarrier flat-fading made 4×4 MIMO SIC/SU-MIMO tractable without a joint space-time processor.
4. Ecosystem momentum: ADSL, 802.11a/g, DVB-T all used OFDM; VLSI toolchains, test equipment, and channel models were already OFDM-native.

Where it falls short: OFDM is sub-optimal for (a) high-mobility vehicular/aerial channels (>500 km/h), (b) UE uplink power efficiency, (c) uncoordinated adjacent-channel coexistence due to OOB, and (d) future THz channels with extreme phase noise. The industry has addressed these with DFT-s-OFDM (UL), PTRS (mmWave), windowed/filtered variants (adjacent-channel), and is actively researching OTFS and AFDM for 6G high-mobility scenarios.

4   Deployment Decision Matrix

The fifth generation of cellular communications is not yet fully deployed, yet the research community and standards bodies are already defining its successor. The ITU-R IMT-2030 framework sets requirements so demanding that they challenge the fundamental assumptions behind CP-OFDM. This section surveys the key technology pillars of 6G: the physics of THz propagation, the delay-Doppler domain waveform OTFS designed for extreme mobility, AI/ML-native waveform design, the integrated sensing-and-communications paradigm, Reconfigurable Intelligent Surfaces, and the green-communications imperative — and asks where, precisely, OFDM and its successors stand in each.

11.1 6G Vision and IMT-2030 Targets (ITU-R WP5D)

ITU-R Working Party 5D published the IMT-2030 Framework Recommendation (M.2160, 2023) establishing the key performance indicators (KPIs) that candidate technologies must meet by 2030. The jump from 5G to 6G is more radical than the 4G-to-5G transition: several KPIs improve by two orders of magnitude, and entirely new capability classes — native sensing and native AI/ML — are added to the requirement set.

The candidate frequency bands reflect the dual imperative of coverage and capacity:

• FR3 / Upper mid-band (7–24 GHz): Best balance of coverage and capacity for macro deployments; likely the primary 6G deployment band. The 7–15 GHz range in particular offers wide contiguous blocks unavailable in sub-6 GHz.
• Sub-THz (100–300 GHz): Ultra-wideband operation enabling Tbps peak rates in dense indoor or backhaul scenarios. Feasibility confirmed in lab; outdoor deployment remains a research challenge.
• THz (>300 GHz): Long-term research horizon. Sub-cm sensing resolution. Fundamental challenges in device technology, phase noise, and molecular absorption.

11.2 THz Band Challenges for OFDM

The THz band offers extreme bandwidth, but its propagation physics stress every assumption that makes CP-OFDM efficient. Understanding these constraints quantitatively is essential for any system designer working on 6G air-interface selection.

Molecular Absorption

Unlike lower-frequency bands where path loss follows a smooth power law, the THz band has sharply frequency-selective molecular absorption windows. Atmospheric O₂ produces a strong absorption peak near 60 GHz (used deliberately for backhaul isolation in WiGig), while H₂O vapour dominates at 183 GHz and 325 GHz. The implication for OFDM is a non-flat channel at the system level — even before multipath — requiring either careful subband selection or absorption-aware water-filling power allocation.

Phase Noise Scaling

Oscillator phase noise power spectral density grows as the square of the carrier frequency. A 300 GHz oscillator with state-of-the-art design exhibits approximately 100× worse phase noise than a comparable 30 GHz oscillator. In OFDM, phase noise causes two degradation modes:

• Common Phase Error (CPE): A time-domain phase rotation that applies identically to all subcarriers in one OFDM symbol — correctable with pilot-based phase tracking.
• Inter-Carrier Interference (ICI): Energy from one subcarrier leaking into adjacent subcarriers — fundamentally degrades SNR and is not correctable by simple pilots. ICI power scales roughly as the ratio of phase noise bandwidth to subcarrier spacing.

Ultra-Wideband Delay Spread

At 100 GHz of instantaneous bandwidth, a modest channel delay spread of \(\tau_{\max} = 50\,\text{ns}\) translates to 5,000 channel taps at the Nyquist rate. The CP duration must cover this delay spread, incurring significant overhead, and the equaliser must handle a very long channel — erasing the simplicity of one-tap per-subcarrier equalisation unless a grouped-subcarrier or OFDM-with-DFT-spreading approach is used.

Extreme Doppler at THz

At a carrier of \(f_c = 300\,\text{GHz}\) and pedestrian speed \(v = 100\,\text{km/h}\), the Doppler shift is:

Path Loss and Pencil Beamforming

Free-space path loss at 300 GHz over 10 m is approximately 102 dB — roughly 20 dB worse than at 30 GHz over the same distance. Atmospheric absorption adds further loss. Compensation requires extremely narrow beams: with \(N_{\text{ant}} = 1024\) elements, the array gain is \(10\log_{10}(1024) \approx 30\,\text{dB}\) — barely sufficient to close the link at short range. Beam management at THz must operate at microsecond timescales, far faster than 5G mmWave beam management.

11.3 OTFS for 6G High-Mobility Scenarios

Orthogonal Time Frequency Space (OTFS), proposed by Hadani et al. (2017), modulates information in the delay-Doppler (DD) domain rather than the time-frequency domain used by OFDM. In channels with extreme Doppler — LEO satellite communications being the canonical example — OTFS achieves near-ideal performance while OFDM's subcarrier orthogonality is destroyed.

LEO Satellite Channel: Motivating Numbers

OTFS Signal Model

OTFS uses a 2D transform — the Symplectic Finite Fourier Transform (SFFT) — to convert the time-frequency plane into the delay-Doppler plane, where the mobile channel appears as a sparse, separable (delay, Doppler) impulse response:

OTFS vs OFDM: Capacity and Diversity

• Ergodic capacity: OTFS and OFDM are theoretically equivalent in ergodic capacity — both achieve the Shannon limit given optimal coding and water-filling power allocation. The key difference is diversity.
• Diversity order: In a channel with \(P\) paths, OTFS achieves full delay-Doppler diversity order \(P\); CP-OFDM with a single-tap equaliser achieves diversity 1 per subcarrier (without coding). Uncoded BER curves differ dramatically.
• Channel estimation overhead: OTFS requires only a single pilot in the DD plane to estimate all \(P\) paths simultaneously (impulse pilot method). CP-OFDM requires pilot tones every \(\sim 14\) symbols in time and every few subcarriers in frequency, scaling with the channel coherence time and bandwidth.
• 3GPP NTN (Non-Terrestrial Networks): Rel-17 and Rel-18 introduced NTN support in NR, including Doppler pre-compensation at the UE. OTFS is actively studied as a candidate for 5G Advanced / 6G NTN air interface (SI: RP-230591).

11.4 AI/ML-Native Waveform Design

The conventional communications system design paradigm — derive an analytical channel model, engineer each block (modulation, channel coding, equalization, detection) separately to match the model — faces fundamental limitations in the complex, non-stationary channels of THz and high-mobility 6G. The emerging alternative is end-to-end learning: treat the transmitter and receiver jointly as an autoencoder and optimise them together via stochastic gradient descent.

Autoencoder Approach (Simonyan/Gruber Framework)

Key empirical findings from DeepSIG, OpenAirInterface (OAI), and the academic literature:

• AWGN channel: The autoencoder converges to constellations that closely resemble standard QAM/PSK — confirming that classical modulation is near-optimal in AWGN, and that the neural network rediscovers this independently.
• Fading channel (known to RX): The autoencoder adapts the constellation geometry and pulse shape to maximise diversity. The learned constellations deviate significantly from rectangular QAM at low SNR.
• Unknown or mismatched channel: End-to-end learning outperforms OFDM with model-based equalization when channel statistics deviate from the assumed model — a crucial advantage in THz channels with rapidly varying absorption and scattering.
• Generalization: A single autoencoder trained on a distribution of channels (e.g., stochastic CDL-A to CDL-E) generalises better than a fixed modulation scheme matched to one channel type.

3GPP AI/ML Channel Estimation — Rel-18 Study Item

3GPP Release 18 includes a study item (RP-213599) on AI/ML for NR physical layer enhancements, covering three use cases: (1) CSI feedback compression, (2) beam management, (3) channel estimation and equalisation. The Rel-18 study is not normative — it defines the evaluation framework and assesses feasibility. Normative AI/ML PHY enhancements are targeted for Rel-19/20.

Semantic Communication

Beyond bit-level optimisation, semantic communication transmits meaning rather than bits. For structured data (images, speech, sensor readings), joint source-channel coding at the semantic level achieves 10–100× compression over separate source coding + channel coding pipelines at the same reconstruction quality. This requires a shared semantic model between transmitter and receiver — a major departure from the bit-pipe model underlying all existing standards.

11.5 ISAC — Integrated Sensing and Communications

ISAC (also called Dual-Function Radar-Communication, DFRC) uses a single waveform, transceiver hardware, and spectrum for both wireless communications and radar-like environmental sensing. OFDM's structure — known pilots on a regular time-frequency grid — makes it a natural dual-purpose waveform.

OFDM Radar Processing

Unified ISAC Signal Model

ISAC Pareto Frontier and Trade-off

The achievable region for simultaneous communications (bit rate \(R\)) and sensing (estimation SINR \(\Gamma_s\)) forms a Pareto frontier parameterised by the power split between data and reference (pilot/sensing) symbols. Key results:

• Dedicating all power to data (standard OFDM) maximises \(R\) but reduces sensing SINR — pilots are sparse and sensing relies on random data correlations.
• Dedicating all power to sensing reference signals maximises \(\Gamma_s\) but \(R = 0\).
• The Pareto-optimal operating point depends on the specific use case. For automotive sensing at 79 GHz, a 10–20% pilot density sacrifice gives >30 dB sensing SINR with <5% capacity loss.

3GPP ISAC in 5G NR and 6G

• SRS-based monostatic sensing (Rel-19): The gNB transmits uplink SRS resources for the UE; in monostatic mode the gNB's own SRS transmission reflects off targets back to the same gNB. WI: NR_ISAC_Ph1 (RP-251881).
• PRS-based bistatic sensing: A reference gNB transmits PRS; a sensing gNB receives the reflected/scattered signal from targets. Uses the existing NR positioning reference signal infrastructure.
• 6G native ISAC: Rather than retrofitting sensing onto a communications waveform, 6G targets dual-function waveform design from the ground up: waveform basis, numerology, pilot pattern, and beamforming jointly optimised for both link performance and sensing accuracy KPIs.

11.6 RIS — Reconfigurable Intelligent Surfaces

A Reconfigurable Intelligent Surface (RIS) is a planar array of passive electromagnetic elements, each with a controllable reflection phase shift. Unlike an active relay, a RIS does not amplify or decode the signal — it passively steers reflected wavefronts. This enables coverage extension without additional RF chains, power amplifiers, or noise amplification.

OFDM + RIS Channel Model

The passive beamforming gain of an \(L\)-element RIS scales as \(L^2\) (coherent combination), compared to \(L\) for active arrays. This quadratic scaling is the primary motivation: a 256-element RIS provides 48 dB of passive beamforming gain, compensating for the 20 dB additional path loss of mmWave / sub-THz bands over microwave. Practical constraints include:

• Channel state information acquisition — the RIS is passive, so the controller must obtain CSI via active sounding through the RIS.
• Phase resolution — 1-bit quantised phase control (0° or 180°) gives approximately 3 dB loss vs continuous phase; 2-bit (4 phases) recovers most of the theoretical gain.
• Coupling and mutual impedance between adjacent elements at sub-wavelength spacing.

11.7 Energy Efficiency in 6G

The 100× bits-per-Joule target relative to 5G is the most challenging KPI in the IMT-2030 framework. The global ICT sector currently consumes approximately 2–3% of world electricity; with the projected 100× traffic growth from 5G to 6G, aggressive energy efficiency improvement is essential for sustainability.

OFDM Energy Overhead

CP-OFDM has inherent energy inefficiencies compared to narrowband alternatives:

• High PAPR (Peak-to-Average Power Ratio): OFDM signals with \(N\) subcarriers have PAPR growing as \(\log N\). Power amplifiers must be backed off from peak power, dramatically reducing their power-added efficiency (PAE). A typical 5G massive MIMO base station PA operates at 5–10% PAE due to OFDM PAPR; at 100% efficiency it would use 10–20× less energy for the same transmitted power.
• CP overhead: \(\mu = N_{\text{CP}} / (N + N_{\text{CP}})\) of all transmitted energy is discarded at the receiver.
• FFT processing energy: At 100 GHz bandwidth, the ADC, FFT, and baseband processing power is dominated by the FFT at \(O(N\log N)\) operations per OFDM symbol, at clock speeds approaching \(10^{11}\) operations/second.

Green 6G Alternatives

• Wake-up receivers: Secondary ultra-low-power receiver (1–10 μW) keeps the primary radio off until a wake-up signal is received, enabling near-zero idle power IoT devices.
• Ambient backscatter communication: Sensors modulate data onto reflected ambient RF energy (from existing 5G/WiFi signals) without any local oscillator or power amplifier. Achieves μW-level operation — incompatible with OFDM but viable for simple IoT sensor reporting at very low data rates.
• Dynamic spectrum access: Cognitive-radio-style spectrum sharing reduces the power consumed in idle monitoring of spectrum.
• Near-zero-energy IoT: Energy harvesting from RF, light, or mechanical vibration, combined with ultra-narrowband (UNB) modulation — the opposite end of the design space from wideband OFDM.

11.8 Summary: OFDM Evolution Across Generations

This section walks through the normative OFDM baseband signal models defined in 3GPP TS 36.211 (LTE) and TS 38.211 (NR), covering downlink CP-OFDM, uplink DFT-s-OFDM (SC-FDMA in LTE), and the Sounding Reference Signal (SRS) for both generations. Each model is accompanied by an interactive chart.

12.1 LTE Physical Layer — Parametric Overview TS 36.211 §5–6

LTE uses CP-OFDM on the downlink (PDSCH, PDCCH, etc.) and DFT-s-OFDM (SC-FDMA) on the uplink (PUSCH) to keep the UE power-amplifier back-off low. Key parameters are fixed across all bandwidths:

Subcarrier spacing is fixed at \(\Delta f = 15\,\text{kHz}\), yielding a useful symbol duration \(T_u = 1/\Delta f = 66.67\,\mu\text{s}\). The first CP of each 0.5 ms slot is slightly longer (5.21 μs) to maintain the 0.5 ms frame alignment.

12.2 LTE Downlink — CP-OFDM Signal Model TS 36.211 §6.12

The complex baseband OFDM time-domain signal for OFDM symbol \(\ell\) in subframe \(s\) is (TS 36.211 Eq. 6.12-1):

where \(a_{k,\ell}\) is the complex modulation symbol on subcarrier \(k\), symbol \(\ell\); \(T_s = 1/(15000 \times 2048) \approx 32.55\,\text{ns}\) is the basic LTE time unit; \(N_{\rm CP}\) is the CP length in samples (160 for the first symbol, 144 for symbols 1–6 in a 0.5 ms slot). Implemented as IFFT + CP prepend:

The resource grid spans \(N_{\rm RB}\times 12\) subcarriers in frequency and 14 OFDM symbols per 1 ms subframe (normal CP).

12.3 LTE Uplink — DFT-s-OFDM (SC-FDMA) TS 36.211 §5.3–5.4

To reduce PAPR and relax the UE power amplifier, the LTE uplink uses DFT-spread OFDM. The transmit chain (TS 36.211 §5.3) is:

The DFT step spreads each QAM symbol across all \(M\) subcarriers allocated to the UE, creating a single-carrier signal in the time domain with PAPR comparable to QPSK even for higher-order modulations. \(M\) must be a product of 2, 3, and 5 (e.g.\ 12, 24, 36, …, 1200) — TS 36.211 Table 5.3-1.

Localised (LFDMA) and interleaved (IFDMA) subcarrier mappings are supported. LTE uses exclusively LFDMA (contiguous allocation):

The received signal after flat-fading channel \(H_k\) and OFDM demodulation:

A single-tap frequency-domain equalizer is then applied before the M-pt IDFT to recover \(\hat{d}_m\).

12.4 LTE Sounding Reference Signal (SRS) TS 36.211 §5.5.3

The SRS is a UL reference signal used by the eNB for uplink channel sounding (wideband CSI acquisition, scheduling, beamforming). Key properties (TS 36.211 §5.5.3):

• Base sequence: Zadoff-Chu (ZC) of length \(M_{\rm SRS}\), cyclic-extended to fill the allocated bandwidth.
• Sequence: \(r^{(\alpha)}_u(n) = e^{j\alpha n}\,\bar{r}_u(n)\), where \(\bar{r}_u\) is the ZC root sequence for group \(u\) and \(\alpha = 2\pi n_{\rm cs}/12\) is the cyclic shift.
• Bandwidth: \(M_{\rm SRS} \in \{4,8,12,\ldots\} \times N_{\rm RB}\) subcarriers; up to 4 nested hopping bandwidths \(m_{\rm SRS,0..3}\).
• Subframe mapping: occupies the last OFDM symbol of the subframe; cannot coexist with PUCCH in same RBs.
• Frequency hopping: controlled by \(b_{\rm hop}\), allowing periodic wideband sounding via interleaved narrow allocations.

12.5 5G NR Physical Layer — Flexible Numerology TS 38.211 §4–5

5G NR uses CP-OFDM for both DL and UL, with DFT-s-OFDM as an optional UL mode. The key flexibility is numerology \(\mu\): subcarrier spacing scales as \(\Delta f = 2^\mu \times 15\,\text{kHz}\), TS 38.211 Table 4.3.2-1.

All numerologies share the same 10 ms frame and 1 ms subframe boundary, enabling mixed-numerology operation within a single carrier (BWP switching). The slot duration is \(T_{\rm slot} = 14 \times (T_u + T_{\rm CP})\) for normal CP.

12.6 5G NR Downlink — CP-OFDM Signal Model TS 38.211 §7.4

The NR DL baseband signal is (TS 38.211 Eq. 7.4-1):

where \(T_c = 1/(\Delta f_{\rm ref}\cdot N^{\rm ref}_{\rm FFT}) = 1/(480\,\text{kHz}\times 4096) \approx 0.509\,\text{ns}\) is the NR basic time unit; \(\Delta f_\mu = 2^\mu \times 15\,\text{kHz}\); and \(N^\mu_{\rm FFT}\) scales with bandwidth class and numerology.

Typical NR FFT sizes (100 MHz FR1, μ=1, Δf=30 kHz):

The NR resource grid spans up to 3300 subcarriers (275 RBs × 12) for 100 MHz / SCS 30 kHz — significantly wider than LTE, reflecting NR's flexible bandwidth part (BWP) architecture.

12.7 5G NR Uplink — CP-OFDM and DFT-s-OFDM TS 38.211 §6.3

Unlike LTE, NR supports both CP-OFDM and DFT-s-OFDM on the uplink (TS 38.211 §6.3). The UE capability bit transformPrecoding (TS 38.214 §6.1.3) selects between the two:

• CP-OFDM UL (default): same structure as DL; enables MIMO spatial multiplexing and supports up to 4 layers. Used when PA back-off is less critical (power class 3, FR2).
• DFT-s-OFDM UL (transform precoding enabled): same as LTE SC-FDMA; lower PAPR, better coverage at cell edge. Mandatory for power class 3 UL coverage enhancement; mandatory for PRACH.

The NR DFT-s-OFDM chain (TS 38.211 §6.3.1.4):

NR DFT-s-OFDM adds a key enhancement: spectral shaping (π/2-BPSK) reduces PAPR further to ~0 dB overhead, enabling 5G to operate in extreme coverage scenarios (TS 38.211 §6.3.1.1, NR PUSCH enhancement Rel-16).

12.8 5G NR Sounding Reference Signal (SRS) TS 38.211 §6.4.1.4

NR SRS is significantly enhanced over LTE SRS (TS 38.211 §6.4.1.4):

The NR SRS sequence (TS 38.211 §6.4.1.4.2):

where \(\alpha_p = 2\pi n^{(p)}_{\rm cs}/N_{\rm cs}\) is the cyclic shift for antenna port \(p\); \(\bar{r}_{u,v}\) is a ZC base sequence of group \(u\), base sequence \(v\); and \(N_{\rm cs} \in \{8, 12\}\) (comb 2/4). Mapped to every \(K_{\rm TC}\)-th subcarrier (\(K_{\rm TC} = 2\) or \(4\) for comb-2/4):

12.9 LTE vs NR OFDM Signal Generation — Head-to-Head