Nyquist Criterion

Designing a baseband digital system requires choosing a pulse shape that avoids ISI at the sampling instants. The Nyquist criterion provides the exact mathematical condition a pulse must satisfy for ISI-free detection. Understanding this criterion also directly leads to the raised cosine filter, which is the standard practical pulse shaping solution used in virtually all modern digital communication systems.

The Nyquist ISI Criterion

Harry Nyquist formulated the condition for ISI-free transmission in the 1920s. The criterion states: a pulse p(t) produces no ISI at the sampling instants if and only if p(nT) = 1 for n = 0 and p(nT) = 0 for all nonzero integers n. In the frequency domain, this is equivalent to requiring that the Fourier transform P(f) satisfies the Nyquist condition: the sum of P(f + k/T) for all integers k must equal a constant (usually T) for all f.

The simplest pulse satisfying this condition is the ideal sinc pulse, p(t) = sinc(t/T), whose Fourier transform is a rectangular function of bandwidth exactly 1/(2T). However, the sinc pulse is not realisable in practice because it extends infinitely in time and requires a perfectly sharp rectangular filter, which cannot be implemented.

The solution to this practical problem is the raised cosine (RC) filter, which gradually rolls off the frequency response instead of cutting off sharply. This roll-off makes the filter physically implementable while still satisfying the Nyquist zero-ISI condition at all sampling instants.

Raised Cosine Filter and Roll-Off Factor

The raised cosine filter has a frequency response that transitions smoothly from its passband to zero over a transition band controlled by the roll-off factor r (also called excess bandwidth factor). The parameter r ranges from 0 to 1. When r = 0, the filter reduces to the ideal rectangular (sinc) filter. When r = 1, the transition band occupies the full Nyquist bandwidth, doubling the total bandwidth used.

The bandwidth occupied by a raised cosine filtered signal is B = (1 + r) / (2T), where T is the symbol period. In terms of symbol rate Rs = 1/T, this becomes B = Rs(1 + r)/2. The bandwidth efficiency in bits per second per hertz for binary signaling is 2/(1 + r). When r = 0, efficiency is 2 bps/Hz; when r = 1, it drops to 1 bps/Hz.

In the time domain, the raised cosine pulse has the form p(t) = sinc(t/T) multiplied by a cosine envelope factor. The key property is that p(nT) = 0 for all nonzero n, regardless of the value of r. The zero crossings remain at exact multiples of T, so ISI is zero at every sampling instant.

Square Root Raised Cosine Implementation

In practical systems, the raised cosine filtering is split equally between the transmitter and receiver using square root raised cosine (SRRC) filters. The transmit filter has frequency response equal to the square root of the RC spectrum, and so does the receive filter. The cascade of the two SRRC filters produces the complete raised cosine response, satisfying the Nyquist criterion after matched filtering at the receiver.

This split is important because the receive SRRC filter also acts as a matched filter for the SRRC transmitted pulse, simultaneously maximising SNR and ensuring ISI-free sampling. This dual benefit makes SRRC the industry standard for pulse shaping in systems like 3G, 4G, and cable modems.

Mathematical Expression

The raised cosine frequency response H_RC(f) is defined in three regions. For |f| less than (1-r)/(2T), H_RC(f) = T. For (1-r)/(2T) less than |f| less than (1+r)/(2T), H_RC(f) = (T/2)[1 + cos(pi*T/r * (|f| - (1-r)/(2T)))]. For |f| greater than (1+r)/(2T), H_RC(f) = 0. The occupied bandwidth is B = (1+r)/(2T).

Given:
Symbol rate Rs = 2 Mbaud
Roll-off factor r = 0.5

Why this formula applies:
Bandwidth of raised cosine signal is B = Rs*(1+r)/2
This gives the actual RF or baseband bandwidth needed for transmission.

Formula:
B = Rs * (1 + r) / 2

Substitution:
B = 2,000,000 * (1 + 0.5) / 2
B = 2,000,000 * 1.5 / 2

Calculation:
B = 3,000,000 / 2 = 1,500,000 Hz

Final Answer:
Required bandwidth B = 1.5 MHz
Compare: Nyquist minimum (r=0) would be 1.0 MHz.
Excess bandwidth due to roll-off = 0.5 MHz (50% overhead).

Exam Tip: In GATE problems, if symbol rate Rs and roll-off factor r are given, bandwidth B = Rs*(1+r)/2. Never confuse this with bit rate. For QPSK with Rs = Rb/2, substitute Rs first, then apply the roll-off formula.

• Nyquist criterion in time domain: p(nT) = delta(n), meaning pulse must be zero at all nonzero sampling instants.
• Raised cosine filter extends the ideal sinc spectrum with a smooth cosine transition band parameterised by roll-off factor r.
• Increasing r increases bandwidth but reduces sensitivity to timing errors and filter implementation difficulty.
• SRRC filters are used in practice so both transmit shaping and receive matched filtering objectives are met simultaneously.
• Bandwidth efficiency trades off directly with roll-off factor: higher r means more bandwidth consumed per unit of data rate.

Quick Revision

• Nyquist criterion: pulse must have zero crossings at all nonzero multiples of T for ISI-free detection.
• Ideal pulse: sinc(t/T), occupies minimum bandwidth B = 1/(2T) = Rs/2. Not realisable.
• Raised cosine bandwidth: B = Rs*(1+r)/2 where r is the roll-off factor, 0 <= r <= 1.
• r = 0 gives minimum Nyquist bandwidth; r = 1 doubles the bandwidth.
• SRRC filters are used in transmitter and receiver; their cascade equals the full raised cosine response.
• GATE trap: bandwidth efficiency = 2/(1+r) bps/Hz only for binary. For M-ary, multiply by log2(M).
• All values of r from 0 to 1 satisfy the zero-ISI Nyquist condition at sampling instants.