GATE Signals and Systems formula sheet

249 formulas in 9 topics, each with when to use it. Each is explained in the topic notes.

EVEN AND ODD PARTS

x_{e} (t) = \frac{x ( t ) + x ( - t )}{2}, x_{o} (t) = \frac{x ( t ) - x ( - t )}{2}, x (t) = x_{e} (t) + x_{o} (t)

Every signal splits this way, uniquely; the DT form is identical with $n$ in place of $t$ . Example: $u (t) = \frac{1}{2} + \frac{1}{2} sgn (t)$ , so $u_{e} (t) = \frac{1}{2}$ and $u_{o} (t) = \frac{1}{2} sgn (t)$ for $t \neq = 0$ .

UNIT STEP, SIGNUM AND RAMP

u (t) = {1, 0, t > 0 t < 0, sgn (t) = 2 u (t) - 1, r (t) = t u (t) = \int_{- \infty}^{t} u (τ) d τ

$sgn (0) = 0$ , so $sgn (t) = 2 u (t) - 1$ holds for $t \neq = 0$ . Derivative chain: $\frac{d}{d t} r (t) = u (t)$ , $\frac{d}{d t} u (t) = δ (t)$ , $\frac{d}{d t} sgn (t) = 2 δ (t)$ .

LTI systems: properties, impulse response, convolution and correlation

All 36 formulas

LINEARITY (SUPERPOSITION)

T {a x_{1} + b x_{2}} = a T {x_{1}} + b T {x_{2}}

Must hold for all inputs $x_{1}, x_{2}$ and all constants $a, b$ (complex constants if the signals may be complex). It combines additivity ( $a = b = 1$ ) and homogeneity ( $b = 0$ ). Zero input giving zero output is necessary, not sufficient.

TIME INVARIANCE

x (t) \to y (t) \Rightarrow x (t - t_{0}) \to y (t - t_{0}) for every t_{0}

Discrete time: $x [n - n_{0}] \to y [n - n_{0}]$ for every integer $n_{0}$ . Also called shift invariance.

Fourier series

All 25 formulas

EXPONENTIAL FOURIER SERIES (SYNTHESIS AND ANALYSIS)

x (t) c_{k} = k = - \infty \sum \infty c_{k} e^{j k ω_{0} t} = \frac{1}{T _{0}} \int_{T_{0}} x (t) e^{- j k ω_{0} t} d t

Holds for real or complex $x (t)$ . $c_{0}$ is the average (DC) value. Plotting $∣ c_{k} ∣$ and $∠ c_{k}$ against $k ω_{0}$ gives the two-sided line spectrum.

TRIGONOMETRIC FOURIER SERIES

x (t) = a_{0} + k = 1 \sum \infty (a_{k} cos k ω_{0} t + b_{k} sin k ω_{0} t)

For real $x (t)$ , with $a_{0}$ the DC value (convention above).

Fourier transform, frequency response, group and phase delay

All 34 formulas

FOURIER TRANSFORM PAIR (Ω FORM, USED IN THESE NOTES)

X (ω) = \int_{- \infty}^{\infty} x (t) e^{- j ω t} d t x (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} X (ω) e^{j ω t} d ω

$ω$ in rad/s. The $1/2 π$ sits only in the inverse, so it reappears in duality, the multiplication property, Parseval's theorem and the formula for $x (0)$ .

FOURIER TRANSFORM PAIR (F FORM)

X (f) = \int_{- \infty}^{\infty} x (t) e^{- j 2 π f t} d t x (t) = \int_{- \infty}^{\infty} X (f) e^{j 2 π f t} df

$f$ in Hz; symmetric, with no factor outside the integrals. Convert with $X (f) = X (ω)_{ω = 2 π f}$ , and for impulses use $δ (ω) = δ (f) /2 π$ , so $2 π δ (ω)$ becomes $δ (f)$ .

Laplace transform, poles, zeros and system functions

All 25 formulas

BILATERAL LAPLACE TRANSFORM

X (s) = L {x (t)} = \int_{- \infty}^{\infty} x (t) e^{- s t} d t, s = σ + j ω

Converges only for $s$ in the ROC, which must be quoted with $X (s)$ .

UNILATERAL LAPLACE TRANSFORM

X (s) = \int_{0^{-}}^{\infty} x (t) e^{- s t} d t

Lower limit $0^{-}$ , so an impulse at $t = 0$ is included. It equals the bilateral transform of the part of $x (t)$ from $t = 0^{-}$ onwards, so, when it exists, its ROC is a right half-plane $Re (s) > σ_{0}$ (the whole plane if that part has finite duration) and is usually not written. For a causal signal the two transforms are identical. This is the version used for differential equations with initial conditions.

Sampling and reconstruction

All 18 formulas

IMPULSE TRAIN AND ITS TRANSFORM

n = - \infty \sum \infty δ (t - n T_{s}) \leftrightarrow f_{s} k = - \infty \sum \infty δ (f - k f_{s})

In $ω$ : the transform is $ω_{s} \sum_{k} δ (ω - k ω_{s})$ , with $ω_{s} = 2 π / T_{s}$ .

SPECTRUM OF THE IDEALLY SAMPLED SIGNAL

X_{s} (f) = \frac{1}{T _{s}} k = - \infty \sum \infty X (f - k f_{s}) = n = - \infty \sum \infty x (n T_{s}) e^{- j 2 π f n T_{s}}

Replicas of $X (f)$ at every multiple of $f_{s}$ , each scaled by $1/ T_{s}$ . In $ω$ : $X_{s} (ω) = \frac{1}{T _{s}} \sum_{k} X (ω - k ω_{s})$ — the scale factor is still $1/ T_{s}$ , not $ω_{s}$ .

z-transform and discrete-time systems

All 32 formulas

BILATERAL Z-TRANSFORM

X (z) = Z {x [n]} = n = - \infty \sum \infty x [n] z^{- n}

Defined only for $z$ in the ROC, the set of $z$ where the sum converges absolutely. Always quote $X (z)$ together with its ROC.

UNILATERAL Z-TRANSFORM

X^{+} (z) = n = 0 \sum \infty x [n] z^{- n}

Ignores $x [n]$ for $n < 0$ ; it equals the bilateral transform of $x [n] u [n]$ . When it exists, its ROC is the exterior of a circle, $∣ z ∣ > r$ (including $z = \infty$ ), so it is usually not written. For a causal sequence the two transforms are identical.

DTFT, DFT and FFT

All 33 formulas

DTFT (ANALYSIS EQUATION)

X (e^{j ω}) = n = - \infty \sum \infty x [n] e^{- j ω n}

Bilateral sum over all $n$ , with $ω$ in rad/sample. Defined when the sum converges (see existence below).

INVERSE DTFT (SYNTHESIS EQUATION)

x [n] = \frac{1}{2 π} \int_{- π}^{π} X (e^{j ω}) e^{j ω n} d ω

The integral may be taken over any interval of length $2 π$ . In the $ω$ convention the $1/2 π$ belongs to the inverse.

FIR and IIR filters

All 23 formulas

FIR FILTER OF LENGTH N

y [n] = k = 0 \sum N - 1 b_{k} x [n - k], h [n] = b_{n} (0 \leq n \leq N - 1), H (z) = k = 0 \sum N - 1 h [k] z^{- k}

Order $N - 1$ , taking $h [0] \neq = 0$ and $h [N - 1] \neq = 0$ . Writing $H (z) = (h [0] z^{N - 1} + h [1] z^{N - 2} + \dots + h [N - 1]) / z^{N - 1}$ shows that its only poles are $N - 1$ poles at $z = 0$ . So a causal FIR filter is always BIBO stable.

IIR FILTER AND ITS SIGN CONVENTION

y [n] = k = 0 \sum M b_{k} x [n - k] - k = 1 \sum K a_{k} y [n - k] ⟺ H (z) = \frac{\sum _{k = 0}^{M} b _{k} z ^{- k}}{1 + \sum _{k = 1}^{K} a _{k} z ^{- k}}

Numerator order $M$ , denominator order $K$ . The minus sign in the difference equation puts $+ a_{k}$ in the denominator. So an equation written as $y [n] = a y [n - 1] + x [n]$ gives $H (z) = 1/ (1 - a z^{- 1})$ , with a pole at $z = a$ . The causal filter is stable if and only if every pole left after cancelling common factors satisfies $∣ p_{k} ∣ < 1$ .