Signals, Systems and Communication Engineering — ESE E&T

Weightage: Communication and signal processing are among the heaviest blocks in the E&T papers. The questions are formula-led: modulation index and power, Carson bandwidth, PCM bit rate and Shannon capacity. A short card for each family covers most of them.

1. Signals and systems

A signal is periodic if $x(t+T)=x(t)$, an energy signal if its energy is finite and a power signal if its average power is finite and nonzero. A system is linear if superposition holds and time-invariant if a shift in the input shifts the output equally. It is causal if the output depends on present and past inputs only, and BIBO stable if every bounded input produces a bounded output.

For a linear time-invariant (LTI) system with impulse response $h(t)$, the output is the convolution $y(t)=x(t)\ast h(t)$. In the frequency domain, convolution becomes multiplication: $Y(f)=X(f)H(f)$. The system is causal when $h(t)=0$ for $t<0$ and stable when $h$ is absolutely integrable.

2. Fourier analysis

The Fourier transform moves a signal to the frequency domain. Pairs and properties worth knowing:

• Time scaling: compressing in time expands in frequency.
• Time shift adds a linear phase and leaves magnitude unchanged.
• Modulation: multiplying by $\cos {\omega }_{0}t$ shifts the spectrum to $\pm {\omega }_0$.
• Parseval's theorem: energy in the time domain equals energy in the frequency domain.

The discrete Fourier transform (DFT) of $N$ samples is computed by the FFT in $O(N{\log }_{2}N)$ operations instead of $N^2$.

3. Sampling

The Nyquist theorem says a band-limited signal with highest frequency $f_m$ can be recovered from samples taken at $f_s\ge 2f_m$. Below this rate, spectral copies overlap, an effect called aliasing, and cannot be undone.

Worked example. A signal contains components at 2 kHz and 5 kHz. The Nyquist rate is $2\times 5=10$ kHz.

Real systems use an anti-aliasing filter before the sampler and a reconstruction filter after it.

4. Amplitude modulation

A standard AM wave is $s(t)=A_c\left[1+\mu \cos ({\omega }_{m}t)\right]\cos ({\omega }_{c}t)$ with modulation index $\mu$. Over-modulation ($\mu >1$) distorts the envelope, and an envelope detector then fails.

The power and bandwidth are:

$P_t=P_c\left(1+\frac{{\mu }^2}{2}\right),\eta =\frac{{\mu }^2}{2+{\mu }^2},BW=2f_m$

Worked example. With $P_c=100$ W and $\mu =0.5$, $P_t=100(1+0.125)=112.5$ W, and the efficiency is $0.25/2.25=11.1\%$. Even at $\mu =1$ the useful sideband power is only one-third of the total.

Variants:

• DSB-SC removes the carrier, saving power, but needs a coherent detector.
• SSB removes the carrier and one sideband, with bandwidth $f_m$ and the best power efficiency.
• VSB keeps part of one sideband and is used for television video.

5. Frequency modulation

In FM the instantaneous frequency varies with the message. The modulation index is $\beta =\Delta f/f_m$, with $\Delta f$ the peak deviation. Carson's rule estimates the bandwidth:

$BW\approx 2(\Delta f+f_m)=2f_m(\beta +1)$

Worked example. Commercial FM has $\Delta f=75$ kHz and $f_m=15$ kHz, so $\beta =5$ and $BW=2(75+15)=180$ kHz.

FM has a constant envelope, so it resists amplitude noise, and it provides a capture effect where the stronger of two signals suppresses the other. It pays with a larger bandwidth. Pre-emphasis and de-emphasis counter the rise of noise at high audio frequencies. Narrowband FM ($\beta \ll 1$) has about the same bandwidth as AM.

6. Noise

Thermal noise power is $N=kTB$, with Boltzmann's constant $k$, absolute temperature $T$ and bandwidth $B$. The signal-to-noise ratio (SNR) compares signal and noise power. The noise figure is the SNR at the input divided by the SNR at the output, and the Friis formula shows that the first stage of a cascade dominates the total noise figure, so a low-noise amplifier is placed first.

7. Pulse code modulation and digital modulation

PCM has three steps: sampling, quantisation and encoding. With $n$ bits per sample:

$\text{bit rate}=n{f}_s,SQNR\approx 6.02n+1.76\text{dB}$

Worked example. Telephone speech sampled at 8 kHz with 8 bits gives $8\times 8000=64$ kbps and an SQNR near 50 dB. Each added bit buys 6 dB.

Companding (A-law or $\mu$-law) uses a non-uniform quantiser to give small signals a better SNR. Delta modulation sends one bit per sample, tracking the signal's slope, but suffers from slope overload and granular noise.

Digital carrier modulation:

The BPSK error probability is $P_e=Q\left(\sqrt{2E_b/N_0}\right)$. A matched filter maximises the SNR at the sampling instant. Intersymbol interference is controlled by pulse shaping (raised cosine) that satisfies the Nyquist criterion.

8. Information theory and coding

The entropy of a source is $H=-\sum p_i{\log }_2{p}_i$ bits per symbol.

Worked example. Probabilities $1/2,1/4,1/8,1/8$ give $H=0.5+0.5+0.375+0.375=1.75$ bits.

Shannon's channel capacity for a band-limited channel with Gaussian noise is:

$C=B{\log }_2(1+SNR)$

For $B=3$ kHz and an SNR of 1000 (30 dB), $C=3000{\log }_2(1001)\approx 29.9$ kbps. Reliable communication below $C$ is possible by suitable coding, and impossible above it.

Error control. The Hamming distance between codewords decides performance: a code with minimum distance $d_{min}$ can detect $d_{min}-1$ errors and correct $\lfloor (d_{min}-1)/2\rfloor$. A parity bit gives distance 2, which detects one error. Hamming (7,4) has $d_{min}=3$, so it corrects one error. CRC codes detect burst errors, and convolutional codes with Viterbi decoding are used in mobile systems.

Common traps

• Using $f_s=f_m$. It must be at least $2f_m$.
• Forgetting that AM carries most power in the carrier.
• Leaving the message frequency out of Carson's rule. The bandwidth is twice the sum of deviation and message frequency.
• Applying the SQNR formula without checking full-scale loading.
• Reading Shannon's formula with SNR in dB. Convert to a ratio first.

Memory aids

• "Two B, two F-M, SSB one": bandwidths of AM, DSB, SSB.
• "6 dB a bit": PCM quantisation.
• "Carson: twice deviation plus message": FM bandwidth.

Summary

LTI systems are described by impulse response and frequency response, and Fourier analysis and sampling connect time and frequency. AM trades bandwidth and power efficiency across its variants, while FM gains noise immunity at the cost of bandwidth.

Digital transmission uses PCM and phase, frequency or amplitude keying, and information theory sets the entropy of a source and the capacity of a channel, with codes providing error control.

Exam protocol

• Write the modulation formulas before substituting numbers.
• Convert decibels to ratios before using Shannon's formula.
• Check the sampling rate before the aliasing argument.
• Note whether the detector is coherent or an envelope detector.