GATE EC Communications Formula Sheet

Organised by the official GATE 2027 EC syllabus, Section 7: Communications. Each worked example was checked in code. Practise these: GATE EC Communications previous year questions.

Random processes

Syllabus: Autocorrelation and power spectral density, properties of white noise, filtering of random signals through LTI systems.

Wiener-Khinchin theorem

S_X(f) = \int_{-\infty}^{\infty} R_X(\tau)\, e^{-j2\pi f\tau}\, d\tau, \qquad P = R_X(0) = \int S_X(f)\, df

\begin{gathered} S_X(f) = \int_{-\infty}^{\infty} R_X(\tau)\, e^{-j2\pi f\tau}\, d\tau \\ P = R_X(0) = \int S_X(f)\, df \end{gathered}

Symbols

$R_X(\tau)$ : autocorrelation of a wide-sense stationary process
$S_X(f)$ : power spectral density (W/Hz)

Applies when

Wide-sense stationary processes. A valid PSD is real, even and never negative.

Common mistake

Forgetting that the total power is R(0), the autocorrelation at zero lag.

Worked example

R_{X}

(τ) = e^(−|τ|):

S_{X}

at f = 0: 2 W/Hz (S = 2/(1 + 4π²f²))

Random signals through LTI systems

S_Y(f) = |H(f)|^2 S_X(f), \qquad \text{white input: } P_Y = \frac{N_0}{2}\int |H(f)|^2 df = \frac{N_0}{2}\int h^2(t)\, dt

\begin{aligned} S_Y(f) &= |H(f)|^2 S_X(f), \\ &\text{white input: } P_Y \\ &\quad = \frac{N_0}{2}\int |H(f)|^2 df \\ &\quad = \frac{N_0}{2}\int h^2(t)\, dt \end{aligned}

Symbols

$N_0/2$ : two-sided PSD of the white noise (W/Hz)
$h(t)$ : impulse response

Applies when

LTI system, wide-sense stationary input. The output mean is the input mean times H(0).

Common mistake

Multiplying by |H| instead of |H|², or mixing one-sided N₀ with two-sided N₀/2.

Worked example

White noise

N_{0}

/2 = 10⁻³ W/Hz through an RC low-pass with RC = 1 ms: 0.5 W (N₀/(4RC))

Analog communications

Syllabus: Amplitude modulation and demodulation, angle modulation and demodulation, spectra of AM and FM, super heterodyne receivers.

AM power and efficiency

s(t) = A_c[1 + \mu\cos\omega_m t]\cos\omega_c t, \qquad P_t = P_c\left(1 + \frac{\mu^2}{2}\right), \qquad \eta = \frac{\mu^2}{2 + \mu^2}

\begin{gathered} s(t) = A_c[1 + \mu\cos\omega_m t] \\ \times \cos\omega_c t \\ P_t = P_c\left(1 + \frac{\mu^2}{2}\right) \\ \eta = \frac{\mu^2}{2 + \mu^2} \end{gathered}

Symbols

$\mu$ : modulation index, 0 to 1 for envelope detection
$P_c$ : carrier power
$\eta$ : fraction of power in the sidebands

Applies when

Single-tone AM. With several tones, μ² is the sum of the squares of each index.

Common mistake

Expecting half the power in the sidebands at μ = 1: it is one third at most.

Worked example

μ = 1: 33.3 % efficiency

Modulation index from the envelope

\mu = \frac{A_{max} - A_{min}}{A_{max} + A_{min}}

Symbols

$A_{max}, A_{min}$ : largest and smallest envelope amplitudes

Applies when

Single-tone AM with μ ≤ 1, read from the envelope on an oscilloscope.

Common mistake

Dividing by

A_{max}

alone.

Worked example

Envelope between 5 V and 15 V: 0.5

FM bandwidth (Carson's rule)

\beta = \frac{\Delta f}{f_m}, \qquad B_T \approx 2(\Delta f + f_m) = 2 f_m(\beta + 1)

\begin{aligned} \beta &= \frac{\Delta f}{f_m}, \\ B_T &\approx 2(\Delta f + f_m) \\ &= 2 f_m(\beta + 1) \end{aligned}

Symbols

$\Delta f$ : peak frequency deviation (Hz)
$f_m$ : highest message frequency (Hz)
$\beta$ : modulation index

Applies when

Holds about 98% of the power. Narrowband FM (β ≪ 1) needs about

2f_m

, like AM.

Common mistake

Using 2Δf alone, which leaves out the message bandwidth.

Worked example

Δf = 75 kHz,

f_{m}

= 15 kHz: 180 kHz

Superheterodyne image frequency

f_{LO} = f_{RF} + f_{IF}, \qquad f_{image} = f_{RF} + 2f_{IF}

\begin{aligned} f_{LO} &= f_{RF} + f_{IF}, \\ f_{image} &= f_{RF} + 2f_{IF} \end{aligned}

Symbols

$f_{IF}$ : intermediate frequency
$f_{LO}$ : local oscillator frequency

Applies when

High-side injection (LO above the signal), as in AM broadcast receivers.

Common mistake

Putting the image at

f_{RF}

f_{IF}

, which is the LO itself.

Worked example

f_{RF}

= 1000 kHz,

f_{IF}

= 455 kHz: 1910 kHz

Information theory

Syllabus: Entropy, source coding, mutual information and channel capacity theorem.

Entropy

H(X) = -\sum_i p_i \log_2 p_i\ \ \text{bits/symbol}, \qquad H \le \log_2 M

\begin{gathered} H(X) = -\sum_i p_i \log_2 p_i \\ \text{(bits/symbol)} \\ H \le \log_2 M \end{gathered}

Symbols

$p_i$ : symbol probabilities
$M$ : number of symbols

Applies when

Discrete memoryless source. The maximum, log₂ M, is reached when all symbols are equally likely; source coding can't average fewer bits than H.

Common mistake

Using ln instead of log₂, which gives nats, not bits.

Worked example

Probabilities 1/2, 1/4, 1/4: 1.5 bits/symbol

Shannon-Hartley capacity

C = B\log_2\!\left(1 + \frac{S}{N}\right)\ \text{bit/s}

Symbols

$B$ : channel bandwidth (Hz)
$S/N$ : signal-to-noise power ratio, as a ratio not in dB

Applies when

Band-limited channel with additive white Gaussian noise. Error-free transmission is possible at any rate below C.

Common mistake

Putting the SNR in dB into the formula: convert 30 dB to 1000 first.

Worked example

B = 3 kHz, SNR = 30 dB: 29.9 kbit/s

Binary symmetric channel capacity

C = 1 - H_b(p), \qquad H_b(p) = -p\log_2 p - (1 - p)\log_2(1 - p), \qquad I(X;Y) = H(Y) - H(Y \mid X)

\begin{gathered} C = 1 - H_b(p) \\ H_b(p) = -p\log_2 p \\ \quad - (1 - p)\log_2(1 - p) \\ I(X;Y) = H(Y) - H(Y \mid X) \end{gathered}

Symbols

$p$ : crossover (bit error) probability
$I(X;Y)$ : mutual information

Applies when

Capacity is the largest mutual information over input distributions; for the BSC, equally likely inputs reach it.

Common mistake

Thinking capacity falls to 0 at p = 1: a channel that always flips the bit is perfect once you invert the output.

Worked example

p = 0.1: 0.531 bit per use

Digital communications

Syllabus: PCM, DPCM, digital modulation schemes (ASK, PSK, FSK, QAM), bandwidth, inter-symbol interference, MAP, ML detection, matched filter receiver, SNR and BER. Fundamentals of error correction, Hamming codes, CRC.

PCM bit rate

R_b = n f_s, \qquad B_{min} = \frac{R_b}{2}, \qquad \text{SQNR} = 6.02n + 1.76\ \text{dB}

\begin{aligned} R_b &= n f_s, \\ B_{min} &= \frac{R_b}{2}, \\ \text{SQNR} &= 6.02n + 1.76\ \text{dB} \end{aligned}

Symbols

$n$ : bits per sample
$f_s$ : sampling rate (samples/s)

Applies when

Uniform quantiser, full-scale sine for the SQNR;

B_{min}

is the Nyquist bandwidth for binary signalling.

Common mistake

Sampling at the message bandwidth instead of at least twice it.

Worked example

Speech sampled at 8 kHz with 8 bits: 64 kbit/s

Matched filter

h(t) = s(T - t), \qquad \text{SNR}_{max} = \frac{2E}{N_0}

\begin{aligned} h(t) &= s(T - t), \\ \text{SNR}_{max} &= \frac{2E}{N_0} \end{aligned}

Symbols

$s(t)$ : the known pulse, of duration T
$E$ : pulse energy (J)
$N_0/2$ : two-sided noise PSD

Applies when

Additive white noise; the output is sampled at t = T. The peak SNR depends only on the energy, not the pulse shape.

Common mistake

Using E/N₀: with a two-sided PSD of N₀/2 the maximum is 2E/N₀.

Worked example

N_{0}

= 5: 10 (10 dB)

Bit error rates

\text{BPSK: } P_b = Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right), \qquad \text{coherent BFSK: } P_b = Q\!\left(\sqrt{\frac{E_b}{N_0}}\right)

\begin{aligned} \text{BPSK: } P_b &= Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right), \\ &\text{coherent BFSK: } P_b \\ &\quad = Q\!\left(\sqrt{\frac{E_b}{N_0}}\right) \end{aligned}

Symbols

$E_b/N_0$ : energy per bit over noise PSD
$Q(x)$ : Gaussian tail probability

Applies when

AWGN, coherent detection, equally likely bits. BFSK needs 3 dB more

E_{b}

N_{0}

than BPSK for the same error rate.

Common mistake

Putting

E_{b}

N_{0}

into the formula in dB.

Worked example

BPSK at

E_{b}

N_{0}

= 9.6 dB: 0.0000097 (9.7 × 10⁻⁶, about 10⁻⁵)

Raised-cosine bandwidth

B = \frac{R_s}{2}(1 + \alpha), \qquad p(nT) = 0 \text{ for } n \neq 0

\begin{aligned} B &= \frac{R_s}{2}(1 + \alpha), \\ p(nT) &= 0 \text{ for } n \\ &\neq 0 \end{aligned}

Symbols

$R_s$ : symbol rate (symbols/s)
$\alpha$ : roll-off factor, 0 to 1

Applies when

Nyquist pulses: zero ISI at the sampling instants. α = 0 is the ideal sinc with B =

R_{s}

/2.

Common mistake

Using the bit rate for M-ary signalling: the bandwidth depends on the symbol rate,

R_{b}

/log₂M.

Worked example

R_{s}

= 10 kbaud, α = 0.5: 7.5 kHz

MAP decision threshold

\text{choose } {+A} \text{ if } r > \gamma = \frac{\sigma^2}{2A}\ln\frac{P_0}{P_1}

\begin{aligned} &\text{choose } {+A} \text{ if } r \\ &\quad > \gamma \\ &\quad = \frac{\sigma^2}{2A}\ln\frac{P_0}{P_1} \end{aligned}

Symbols

$\pm A$ : the two received levels, sent with probabilities $P_{1}$ (for +A) and $P_{0}$ (for −A)
$\sigma^2$ : noise variance

Applies when

Binary antipodal signals in Gaussian noise. With equal priors MAP becomes ML and the threshold is 0.

Common mistake

Moving the threshold towards the more likely symbol: it moves away from it, enlarging that symbol's region.

Worked example

A = 1, σ² = 0.5,

P_{0}

= 0.75,

P_{1}

= 0.25: 0.275

Hamming codes

2^r \ge m + r + 1, \qquad n = m + r, \qquad d_{min} = 3 \Rightarrow \text{corrects 1 error, detects 2}

\begin{gathered} 2^r \ge m + r + 1 \\ n = m + r \\ d_{min} = 3: \\ \text{corrects 1 error, detects 2} \end{gathered}

Symbols

$m$ : data bits
$r$ : parity bits
$d_{min}$ : minimum Hamming distance

Applies when

Single-error-correcting Hamming codes, such as (7, 4). A code with

d_{min}

corrects ⌊(

d_{min}

− 1)/2⌋ errors.

Common mistake

Using 2^r ≥ m + r, which leaves no syndrome value for 'no error'.

Worked example

4 data bits: 3 parity bits (the (7, 4) code)

CRC remainder

\text{transmit } M(x)\,x^r + R(x), \qquad R(x) = M(x)\,x^r \bmod G(x)\ \ (\text{mod-2 arithmetic})

\begin{gathered} \text{transmit } M(x)\,x^r + R(x) \\ R(x) = M(x)\,x^r \bmod G(x) \\ \text{(mod-2 arithmetic)} \end{gathered}

Symbols

$G(x)$ : generator polynomial of degree r
$M(x)$ : message polynomial

Applies when

Division uses XOR, with no carries. The receiver divides the whole frame by G(x) and gets 0 if no error is detected.

Common mistake

Appending r − 1 zeros instead of r (the degree of G).

Worked example

M = 1101, G = 1011 (x³ + x + 1): the remainder, as a number: 1 (remainder 001; frame 1101001)

Practise these: GATE EC Communications previous year questions

Source for the section headings: the official GATE 2027 EC syllabus, IIT Madras. The formulas, examples and notes are Studyloaf's own.

GATE EC Communications Formula Sheet

Random processes

Wiener-Khinchin theorem

Random signals through LTI systems

Analog communications

AM power and efficiency

Modulation index from the envelope

FM bandwidth (Carson's rule)

Superheterodyne image frequency

Information theory

Entropy

Shannon-Hartley capacity

Binary symmetric channel capacity

Digital communications

PCM bit rate

Matched filter

Bit error rates

Raised-cosine bandwidth

MAP decision threshold

Hamming codes

CRC remainder

Other GATE EC formula sheets