Ferkans — Interactive Telecom Tutor

Why Pulse Shaping Matters

So far we have represented each symbol as a point in signal space, ignoring the pulse shape used to transmit it. In practice, the transmitted waveform is

$x(t) = \sum_k a_k\, p(t - kT_s)$

where $a_k$ are the constellation symbols and $p(t)$ is the transmit pulse shape. The choice of $p(t)$ determines the signal bandwidth and whether neighbouring symbols interfere with each other — the phenomenon of inter-symbol interference (ISI). This section develops the theory of ISI-free pulse shaping.

Definition:
Inter-Symbol Interference (ISI)

Inter-symbol interference (ISI) occurs when the received sample at time $t = kT_s$ contains contributions from symbols other than $a_k$ :

$y(kT_s) = a_k\, p(0) + \underbrace{\sum_{l \neq k} a_l\, p((k-l)T_s)}_{\text{ISI}} + w(kT_s)$

where $p(t)$ is the overall pulse shape (convolution of transmit pulse, channel, and receive filter).

ISI is absent if and only if

$p(nT_s) = \begin{cases} 1 & n = 0 \\ 0 & n \neq 0 \end{cases}$

which is the Nyquist ISI-free condition in the time domain.

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Theorem: Nyquist ISI-Free Criterion (First Nyquist Criterion)

A pulse $p(t)$ with Fourier transform $P(f)$ satisfies the ISI-free condition $p(nT_s) = \delta[n]$ if and only if

$\sum_{k=-\infty}^{\infty} P\!\left(f - \frac{k}{T_s}\right) = T_s, \qquad \forall\, f$

That is, the periodically repeated spectrum $P(f)$ folds into a constant. This is equivalent to the vestigial symmetry condition: $P(f)$ must have odd symmetry about the Nyquist frequency $f_N = 1/(2T_s)$ .

The minimum bandwidth for ISI-free transmission at symbol rate $R_s = 1/T_s$ is $W_{\min} = R_s/2 = 1/(2T_s)$ , achieved by the ideal (sinc) pulse $p(t) = \operatorname{sinc}(t/T_s)$ .

Think of $P(f)$ as a jigsaw puzzle: when you tile the frequency axis with copies of $P(f)$ shifted by multiples of $1/T_s$ , the pieces must fill the frequency axis to a uniform height. The sinc pulse is a single rectangular piece that tiles perfectly. The raised-cosine pulse rounds the edges but maintains the tiling property through symmetry.

Proof

Poisson summation formula

The ISI-free condition $p(nT_s) = \delta[n]$ means the sampled pulse is a unit impulse. By the Poisson summation formula:

$\sum_{n=-\infty}^{\infty} p(nT_s)\, e^{-j2\pi f n T_s} = \frac{1}{T_s} \sum_{k=-\infty}^{\infty} P\!\left(f - \frac{k}{T_s}\right)$

Apply the ISI-free condition

The left side equals $\sum_n \delta[n]\, e^{-j2\pi f n T_s} = 1$ for all $f$ . Therefore:

$\frac{1}{T_s} \sum_{k} P\!\left(f - \frac{k}{T_s}\right) = 1$

which gives the Nyquist criterion. $\blacksquare$

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Definition:
Raised-Cosine Filter

The raised-cosine (RC) filter is the most widely used Nyquist pulse. Its frequency response is

$P_{\text{RC}}(f) = \begin{cases} T_s & |f| \leq \frac{1-\alpha}{2T_s} \\[6pt] \frac{T_s}{2}\left[1 + \cos\!\left(\frac{\pi T_s}{\alpha}\left(|f| - \frac{1-\alpha}{2T_s}\right)\right)\right] & \frac{1-\alpha}{2T_s} < |f| \leq \frac{1+\alpha}{2T_s} \\[6pt] 0 & |f| > \frac{1+\alpha}{2T_s} \end{cases}$

where $\alpha \in [0, 1]$ is the roll-off factor. The time-domain impulse response is

$p_{\text{RC}}(t) = \operatorname{sinc}\!\left(\frac{t}{T_s}\right) \cdot \frac{\cos(\pi\alpha t/T_s)}{1 - (2\alpha t/T_s)^2}$

Key properties:

Bandwidth: $W = (1+\alpha)/(2T_s) = (1+\alpha)R_s/2$
Excess bandwidth: $\alpha \cdot R_s/2$ beyond the Nyquist minimum
$\alpha = 0$ : ideal sinc pulse (minimum bandwidth, slowest decay)
$\alpha = 1$ : widest bandwidth, fastest time-domain decay ( $\sim 1/t^3$ )

Larger $\alpha$ means more bandwidth but better timing robustness (faster tail decay). Practical systems use $\alpha = 0.15$ to $0.35$ .

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Nyquist Pulse Shaping — Roll-off Sweep

Watch how the raised-cosine pulse shape changes as the roll-off factor

\alpha

sweeps from 0 (sinc pulse) to 1 (widest bandwidth). All pulses pass through zero at integer multiples of

T_s

— the ISI-free property — regardless of

\alpha

. Observe the trade-off: larger

\alpha

gives faster tail decay (more robust to timing jitter) at the cost of wider bandwidth.

Raised-cosine pulses for

\alpha = 0, 0.25, 0.5, 0.75, 1.0

. All satisfy the Nyquist ISI-free condition.

Raised-Cosine Pulse and Eye Diagram

Explore how the roll-off factor $\alpha$ affects the pulse shape, spectrum, and eye diagram. Observe that all raised-cosine pulses pass through zero at integer multiples of $T_s$ (the ISI-free property) regardless of $\alpha$ .

Parameters

Roll-off factor

\alpha

0.35

Number of symbols10

Show eye diagram

Eye Diagram for Digital Modulation

The eye diagram overlays multiple symbol periods of the received signal to visualise ISI, noise margin, and timing sensitivity. A "wide open" eye indicates good ISI-free performance; a closed eye indicates severe ISI or noise.

Parameters

Modulation

Roll-off factor

\alpha

0.25

SNR (dB)20

Number of symbols200

Definition:
Root-Raised-Cosine (RRC) Filter

The root-raised-cosine (RRC) filter has frequency response

$P_{\text{RRC}}(f) = \sqrt{P_{\text{RC}}(f)}$

The time-domain impulse response is

$p_{\text{RRC}}(t) = \frac{1}{T_s} \cdot \frac{\sin\!\left(\pi(1-\alpha)t/T_s\right) + 4\alpha t/T_s \cdot \cos\!\left(\pi(1+\alpha)t/T_s\right)}{\pi t/T_s \left[1 - (4\alpha t/T_s)^2\right]}$

The RRC pulse does not satisfy the Nyquist criterion by itself ( $p_{\text{RRC}}(nT_s) \neq \delta[n]$ for $n \neq 0$ ). However, when used at both transmitter and receiver, the cascade produces a raised-cosine pulse, which is ISI-free.

The RRC filter is a matched filter: if the transmit filter is $p_{\text{RRC}}(t)$ and the receive filter is $p_{\text{RRC}}^*(-t)$ , the cascade is $p_{\text{RRC}} * p_{\text{RRC}} = p_{\text{RC}}$ .

Example: Pulse Shaping in LTE and 5G NR

LTE and 5G NR use OFDM, which inherently provides rectangular pulse shaping per subcarrier. However, filtering is applied at the system bandwidth edges.

(a) Why does OFDM not use RRC pulse shaping on each subcarrier?

(b) Where does the RRC concept appear in modern systems?

(c) What is the spectral efficiency impact?

Solution

OFDM subcarrier shaping

Each OFDM subcarrier uses a rectangular window of duration $T_u + T_{\text{CP}}$ in the time domain, giving a sinc-like spectrum per subcarrier. The orthogonality of subcarriers is maintained by the DFT/IDFT structure, not by Nyquist pulse shaping.

RRC in single-carrier systems

RRC pulse shaping is used in:

SC-FDMA (DFT-spread OFDM) in LTE uplink
Single-carrier modes in 5G NR
Satellite links (DVB-S2 uses $\alpha = 0.20$ or $0.35$ )
Backhaul microwave links

The 3GPP standard specifies a spectrum emission mask that effectively requires filtering at the band edges, which approximates raised-cosine roll-off.

Spectral efficiency

With roll-off $\alpha$ , the occupied bandwidth is $W = (1+\alpha)R_s/2$ , giving spectral efficiency $\eta = 2\log_2 M / (1+\alpha)$ .

For $\alpha = 0.22$ (DVB-S2X) and 32-APSK: $\eta = 2 \times 5 / 1.22 = 8.2$ bits/s/Hz. $\blacksquare$

Theorem: Matched Filter Pair (TX RRC + RX RRC = Raised Cosine)

Let the transmit pulse be $g_T(t) = p_{\text{RRC}}(t)$ and the receive filter be $g_R(t) = p_{\text{RRC}}^*(-t)$ . Then:

The overall pulse $p(t) = g_T(t) * g_R(t) = p_{\text{RC}}(t)$ satisfies the Nyquist ISI-free condition.
The receive filter $g_R(t)$ is the matched filter for $g_T(t)$ , maximising the output SNR at the sampling instant.
The output SNR at $t = 0$ is $\text{SNR} = 2E_s/N_0$ , which is the maximum achievable SNR for any receive filter.

This split places equal spectral shaping at transmitter and receiver, ensuring both ISI-free detection and optimal noise rejection.

Splitting the Nyquist filter equally between TX and RX is the unique solution that simultaneously achieves ISI-free transmission and matched-filter optimality. Any other split trades one for the other.

Proof

Cascade in frequency domain

$P(f) = G_T(f) \cdot G_R(f) = P_{\text{RRC}}(f) \cdot P_{\text{RRC}}^*(f) = |P_{\text{RRC}}(f)|^2 = P_{\text{RC}}(f)$

since $P_{\text{RRC}}(f)$ is real and non-negative by construction.

Matched filter optimality

The matched filter for a pulse $g_T(t)$ in AWGN is $g_R(t) = g_T^*(T - t)$ . For a symmetric $p_{\text{RRC}}$ , this is $g_R(t) = p_{\text{RRC}}(-t) = p_{\text{RRC}}(t)$ (the RRC pulse is symmetric).

SNR calculation

Output SNR $= \|g_T\|^2 \cdot 2/N_0 = E_s \cdot 2/N_0 = 2E_s/N_0$ . By the matched filter theorem (Chapter 4), no other $g_R(t)$ can exceed this SNR. $\blacksquare$

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Common Mistake: Forgetting the Matched Filter at the Receiver

Mistake:

Designing the transmit pulse shape carefully but using a simple lowpass filter (e.g., a brick-wall filter) at the receiver instead of the matched RRC filter.

Correction:

Both the ISI-free property and the optimal SNR require the matched pair: TX RRC + RX RRC. Using a non-matched receive filter either introduces ISI (if the cascade is not Nyquist) or reduces SNR (if the filter is not matched), or both.

In practice, the receive RRC filter is implemented digitally after ADC sampling, with the filter coefficients stored in firmware.

Common Mistake: The $\alpha = 0$ Sinc Pulse is Impractical

Mistake:

Targeting $\alpha = 0$ (sinc pulse) to minimise bandwidth, expecting that it can be implemented in a real system.

Correction:

The sinc pulse $p(t) = \operatorname{sinc}(t/T_s)$ decays as $1/|t|$ , which means:

It is non-causal and has infinite duration
A small timing error causes significant ISI because the tails decay slowly
Truncation to finite length introduces spectral leakage

Practical systems use $\alpha \geq 0.1$ (often 0.15 to 0.35) to ensure faster tail decay ( $\sim 1/|t|^3$ for $\alpha > 0$ ) and robustness to timing errors.

Quick Check

A system transmits at symbol rate $R_s = 10$ Msymbols/s using a raised-cosine filter with $\alpha = 0.25$ . What is the occupied signal bandwidth?

5 MHz

6.25 MHz

10 MHz

12.5 MHz

Correction:

6.25 MHz

$W = (1+\alpha)R_s/2 = 1.25 \times 10/2 = 6.25$ MHz.

Inter-Symbol Interference (ISI)

Contamination of the current symbol's sample by energy from adjacent symbols, caused by the spreading of pulses beyond one symbol period. ISI degrades detection performance unless equalised or prevented by Nyquist pulse design.

Raised-Cosine Filter

A Nyquist pulse whose spectrum has a cosine roll-off transition band. Parametrised by the roll-off factor $\alpha \in [0, 1]$ , it satisfies the ISI-free condition for all $\alpha$ .

Root-Raised-Cosine (RRC) Filter

A pulse-shaping filter whose spectrum is the square root of the raised-cosine spectrum. When used at both transmitter and receiver, the cascade is a raised-cosine filter — achieving both ISI-free transmission and matched-filter optimality.

⚠️Engineering Note

Timing Jitter and Sample Clock Accuracy

The Nyquist ISI-free condition $p(nT_s) = \delta[n]$ assumes perfect sampling at $t = nT_s$ . In practice, the sample clock has jitter (random timing error $\Delta t$ ) that introduces residual ISI:

$y(nT_s + \Delta t) \approx a_n\, p(\Delta t) + \sum_{l \neq n} a_l\, p((n-l)T_s + \Delta t)$

The ISI power depends on the pulse tail decay rate:

Sinc pulse ( $\alpha = 0$ ): tails decay as $1/|t|$ , making ISI very sensitive to timing jitter. Even 1% of $T_s$ jitter causes significant degradation.
Raised cosine ( $\alpha > 0$ ): tails decay as $1/|t|^3$ , providing substantial robustness. For $\alpha = 0.25$ , timing jitter of 5% of $T_s$ causes less than 0.5 dB SNR penalty.

In modern systems, timing recovery uses a digital PLL (e.g., Gardner or Mueller-Muller algorithm) that tracks the optimal sampling instant. The residual jitter after timing recovery is typically $< 1\%$ of $T_s$ , but at symbol rates above 1 Gbaud, even this requires careful clock distribution design.

For OFDM systems (Chapter 14), timing jitter manifests as common phase error (CPE) and inter-carrier interference (ICI), which are handled differently from single-carrier ISI.

Practical Constraints

•
Sample clock jitter < 1% of Ts for negligible ISI penalty
•
Roll-off alpha >= 0.1 recommended for timing robustness
•
Digital PLL convergence time impacts preamble design

Signal-Space Geometry and Decision Regions — Voronoi decision regions for 16-QAM. Corner points have two nearest neighbours, edge points have three, and interior points have four. The symbol error probability is dominated by the inner points, which have the most nearest neighbours.

Historical Note: Harry Nyquist and ISI-Free Signaling

1928

Harry Nyquist (1889-1976) established the theoretical foundations for ISI-free pulse transmission in his landmark 1928 paper "Certain Topics in Telegraph Transmission Theory." Working at Bell Labs, Nyquist showed that the maximum symbol rate through a channel of bandwidth $W$ Hz is $2W$ symbols per second — the Nyquist rate. His ISI-free criterion, formalised in the frequency domain, remains the basis of all modern pulse-shaping design.

Pulse Shaping and Nyquist Criterion

Why Pulse Shaping Matters

Definition: Inter-Symbol Interference (ISI)

Theorem: Nyquist ISI-Free Criterion (First Nyquist Criterion)

Poisson summation formula

Apply the ISI-free condition

Definition: Raised-Cosine Filter

Nyquist Pulse Shaping — Roll-off Sweep

Raised-Cosine Pulse and Eye Diagram

Parameters

Eye Diagram for Digital Modulation

Parameters

Definition: Root-Raised-Cosine (RRC) Filter

Example: Pulse Shaping in LTE and 5G NR

OFDM subcarrier shaping

RRC in single-carrier systems

Spectral efficiency

Theorem: Matched Filter Pair (TX RRC + RX RRC = Raised Cosine)

Cascade in frequency domain

Matched filter optimality

SNR calculation

Common Mistake: Forgetting the Matched Filter at the Receiver

Common Mistake: The α=0\alpha = 0α=0 Sinc Pulse is Impractical

Quick Check

Inter-Symbol Interference (ISI)

Raised-Cosine Filter

Root-Raised-Cosine (RRC) Filter

Timing Jitter and Sample Clock Accuracy

Signal-Space Geometry and Decision Regions

Historical Note: Harry Nyquist and ISI-Free Signaling

Definition:
Inter-Symbol Interference (ISI)

Definition:
Raised-Cosine Filter

Definition:
Root-Raised-Cosine (RRC) Filter

Common Mistake: The $\alpha = 0$ Sinc Pulse is Impractical