Signals and systems

Electronics and Communication Engineering (B.E./B.Tech) · Signals and communication · free study notes for exams, placements and interviews

In short

A signal is a function carrying information (voice, ECG, voltage, image pixels); a system is anything that transforms one signal into another (amplifier, filter, channel, microcontroller code). This topic fixes the vocabulary and the mathematics - classification of signals, properties of systems, the impulse response, and convolution - on which every later communication and DSP subject is built. It is a full core course in the third or fourth semester of AICTE/CBCS B.E. and B.Tech curricula, a scoring section of GATE EC/EE/IN and IES/ESE, and a standing favourite in BEL, ISRO, DRDO, BARC, C-DAC, Qualcomm, Texas Instruments, Analog Devices, MediaTek, Tejas Networks and Sasken technical rounds. Interviewers use it to check whether a candidate can reason about a block diagram rather than only plug into formulas.

Explained

Signals: definitions and classification

Signal: a single-valued function of one or more independent variables conveying information. One-dimensional (speech, x(t)), two-dimensional (an image, x(m,n)), multichannel (3-lead ECG).
Continuous-time (CT) vs discrete-time (DT): x(t) is defined for every real t; x[n] is defined only at integers n. A DT signal usually arises by sampling a CT one, but may be inherently discrete (daily closing price).
Analog vs digital: analog = continuous in both time and amplitude; digital = discrete in both (quantised). A sampled-but-unquantised signal is a discrete-time analog signal.
Deterministic vs random: deterministic signals have an exact mathematical description; random signals are described only statistically (covered in the random signals note).
Periodic vs aperiodic: x(t) = x(t + T) for all t; the smallest such T > 0 is the fundamental period T₀, and f₀ = 1/T₀, ω₀ = 2π/T₀. For DT, x[n] = x[n + N] with integer N.
Important DT difference: x[n] = cos(ω₀n) is periodic only if ω₀/2π is rational, i.e. ω₀ = 2πm/N with integers m, N. So cos(n) is *not* periodic, though cos(t) is.
Also in DT: frequency is unique only over a 2π range, because cos((ω₀ + 2π)n) = cos(ω₀n). The highest DT frequency is ω = π.
Energy signals vs power signals: energy signals have finite energy and zero average power (all time-limited and all decaying signals); power signals have finite non-zero average power and infinite energy (all periodic signals, unit step, noise). Some signals such as t·u(t) are neither.
Even and odd: x(−t) = x(t) is even, x(−t) = −x(t) is odd. Any signal splits uniquely as x = xₑ + xₒ with xₑ(t) = ½[x(t) + x(−t)], xₒ(t) = ½[x(t) − x(−t)].
Causal signal: x(t) = 0 for t < 0; anticausal if zero for t > 0; otherwise non-causal (two-sided).
Other labels asked: bounded (|x(t)| ≤ M), absolutely integrable, finite-duration, right-sided, hermitian for complex signals (x(−t) = x*(t)).

Standard signals and signal operations

Unit impulse δ(t): zero everywhere except t = 0, with unit area. It is a distribution, not an ordinary function. Properties: ∫δ(t)dt = 1; x(t)δ(t) = x(0)δ(t); ∫x(t)δ(t − t₀)dt = x(t₀) (sifting); δ(at) = δ(t)/|a|; δ(t) = du(t)/dt.
Unit step u(t): 0 for t < 0, 1 for t > 0 (value at 0 conventionally ½ or 1). Ramp r(t) = t·u(t) = ∫u; rectangular pulse = u(t) − u(t − T); signum, triangular and sinc(t) = sin(πt)/(πt) are the other standard ones.
DT counterparts: δ[n] = 1 at n = 0 else 0 (an ordinary sequence, no pathology); u[n] = Σ_{k≤n} δ[k]; δ[n] = u[n] − u[n−1]; x[n]δ[n−k] = x[k]δ[n−k].
Exponentials: real e^{at} (growing a > 0, decaying a < 0), complex e^{jω₀t} (unit magnitude, period 2π/ω₀), DT aⁿu[n] (decays for |a| < 1), and the damped sinusoid e^{−at}cos(ω₀t).
Time shifting: x(t − t₀) delays by t₀ if t₀ > 0 (shift right); x[n + 3] is an advance.
Time scaling: x(at) compresses if |a| > 1, expands if |a| < 1. In DT, decimation x[2n] loses samples and is not invertible; interpolation x[n/2] needs values to be filled in.
Time reversal (folding): x(−t). Amplitude scaling Ax(t); addition and multiplication are sample-by-sample.
Order matters: for x(at − b), shift and scale in a fixed order - write x(a(t − b/a)), i.e. scale by a then delay by b/a; doing it the other way gives a wrong answer and is a classic exam trap.

Systems: classification and property tests

System: an operator y = T{x} mapping input signal to output signal. SISO/MIMO, CT/DT, hybrid (ADC, DAC).
Linearity: satisfies superposition - T{a₁x₁ + a₂x₂} = a₁T{x₁} + a₂T{x₂} (additivity + homogeneity). A linear system must give zero output for zero input, so y(t) = x(t) + 5 is *not* linear. y[n] = x²[n] and y(t) = |x(t)| are non-linear.
Time (shift) invariance: delaying the input only delays the output - if x[n] → y[n], then x[n − k] → y[n − k]. Test: compute the shifted-input response and the shifted output and compare. y[n] = n·x[n], y[n] = x[2n] and y(t) = x(t)cos(ω t) are time-variant.
LTI system: linear and time-invariant - the only class with a complete, simple description (its impulse response).
Causality: output at any instant depends only on present and past inputs. y[n] = x[n] + x[n−1] is causal; y[n] = x[n+1] is not. Any real-time system must be causal; off-line processing (a stored audio file) need not be.
Memory: memoryless (static) if y at t depends only on x at t (a resistor, y = kx); otherwise dynamic (capacitor, accumulator, delay line).
Stability (BIBO): every bounded input produces a bounded output.
Invertibility: distinct inputs give distinct outputs, so an inverse system exists with h * h_inv = δ. Needed for channel equalisation.
Other classes: static/dynamic, lumped/distributed, deterministic/stochastic, passive/active (passive if output energy ≤ input energy).

LTI systems: impulse response and convolution

Impulse response h(t) or h[n]: the output when the input is δ(t) or δ[n], with the system initially at rest. For an LTI system it is a *complete* description.
Derivation outline (DT): write x[n] = Σ_k x[k]δ[n − k] (sifting); by linearity the output is Σ_k x[k]·T{δ[n−k]}; by time invariance T{δ[n−k]} = h[n−k]; hence y[n] = Σ_k x[k]h[n−k]. The CT case follows identically with the integral replacing the sum.
Convolution properties: commutative x*h = h*x; associative (cascade of LTI blocks ⇒ h = h₁*h₂, order immaterial); distributive over addition (parallel blocks ⇒ h = h₁ + h₂); identity x*δ = x; shifting x(t)*δ(t − t₀) = x(t − t₀).
Graphical CT convolution, step by step: (1) sketch x(τ) and h(τ); (2) fold one to get h(−τ); (3) shift it by t; (4) multiply and integrate the overlap; (5) repeat for each range of t where the overlap limits change; (6) assemble the piecewise answer. Convolving two pulses of widths T₁ and T₂ gives a trapezoid (triangle if equal) of total width T₁ + T₂.
DT convolution facts: if x[n] has N₁ samples and h[n] has N₂, then y[n] has N₁ + N₂ − 1 samples; the starting index is the sum of the starting indices; Σy = (Σx)(Σh) - a quick arithmetic check. The sliding-strip or tabulation (multiply-and-add-diagonals) method is fastest in exams.
Step response: s(t) = u(t) * h(t) = ∫₀^t h(τ)dτ, so h(t) = ds/dt; in DT s[n] = Σ_{k≤n} h[k] and h[n] = s[n] − s[n−1].
Eigenfunction property: for an LTI system, e^{st} in gives H(s)e^{st} out and zⁿ gives H(z)zⁿ - this is the doorway to the transform notes; the transforms themselves are treated there.
Correlation: cross-correlation R_xy(τ) = ∫x(t)y(t + τ)dt measures similarity and delay; autocorrelation R_xx(τ) is even, maximum at τ = 0, and R_xx(0) = energy of x. Correlation equals convolution with the time-reversed signal: R_xy(τ) = x(−τ)*y(τ).

Stability, causality and other representations

BIBO condition: an LTI system is stable if and only if its impulse response is absolutely integrable (CT) or absolutely summable (DT).
Sufficiency proof sketch: |y(t)| = |∫h(τ)x(t−τ)dτ| ≤ ∫|h(τ)||x(t−τ)|dτ ≤ M_x∫|h(τ)|dτ, bounded if the integral is finite. Necessity is shown by choosing x(t) = sgn(h(−t)), which makes y(0) = ∫|h| → ∞.
Causality condition for LTI: h(t) = 0 for t < 0 (h[n] = 0 for n < 0); then the convolution limits become 0 to t (or 0 to n).
Marginal cases: h(t) = u(t) (ideal integrator) and h[n] = u[n] (accumulator) are *not* BIBO stable - their impulse responses are bounded but not absolutely integrable/summable.
Differential / difference equation form: Σ aₖ dᵏy/dtᵏ = Σ bₖ dᵏx/dtᵏ; in DT, Σ aₖ y[n−k] = Σ bₖ x[n−k]. Linear constant-coefficient equations with zero initial conditions describe LTI systems.
Total response = zero-input response (from initial conditions) + zero-state response (from the input) = also = natural response (system's own modes) + forced response (shape of the input). The two decompositions are different; keep them apart.
FIR vs IIR (DT): finite impulse response (h has finitely many non-zero samples, always stable, non-recursive) vs infinite (recursive, feedback, can be unstable).
State-space form: ẋ = Ax + Bu, y = Cx + Du gives an internal description; asymptotic (internal) stability depends on the eigenvalues of A, and may differ from BIBO stability when a pole-zero cancellation hides an unstable mode.
Block-diagram elements: adder, multiplier (gain), integrator (CT) or unit delay z⁻¹ (DT) - any LCCDE can be realised from these three.

CT versus DT: what actually differs

Aspect	Continuous time	Discrete time
Independent variable	real t	integer n
Impulse	δ(t), a distribution, infinite "height"	δ[n] = 1, an ordinary sample
Convolution	integral ∫x(τ)h(t−τ)dτ	sum Σx[k]h[n−k]
Sinusoid periodicity	always periodic, T = 2π/ω₀	only if ω₀/2π is rational
Frequency range	−∞ < ω < ∞, all distinct	unique over 2π, highest at ω = π
Time scaling	x(at) always defined	x[2n] loses data, x[n/2] undefined between samples
Stability test	∫	h(t)	dt < ∞	Σ	h[n]	< ∞
Realisation	R, L, C, op-amps	adders, multipliers, delays in DSP/FPGA

Key points

Signal - a function carrying information; system - an operator mapping input signals to output signals.
Energy signal - finite E, zero average power; power signal - finite non-zero P, infinite energy; periodic signals are power signals.
Average power of a sinusoid - A²/2 regardless of frequency or phase; of A·cos + DC term, powers add.
DT sinusoid periodicity - needs ω₀ = 2πm/N; fundamental N = 2πm/ω₀ for the smallest integer m making N an integer.
Sum of two periodic signals - periodic only if T₁/T₂ is rational; the period is the LCM of T₁ and T₂.
Impulse - δ(t) = du/dt, unit area, sifting property picks out x(t₀); δ[n] = u[n] − u[n−1].
Linearity - superposition; any additive constant, squaring, or |·| destroys it.
Time invariance - a shift in input causes only an equal shift in output; coefficients that depend on n or t destroy it.
LTI system - fully characterised by h(t) or h[n]; output = input convolved with h.
Convolution - commutative, associative, distributive; cascade ⇒ h₁*h₂, parallel ⇒ h₁+h₂.
DT convolution length - N₁ + N₂ − 1 samples, and Σy = Σx · Σh as an arithmetic check.
Step response - integral (or running sum) of the impulse response; h is its derivative (or first difference).
BIBO stability - ∫|h|dt < ∞ or Σ|h[n]| < ∞; an integrator and an accumulator fail this test.
Causality of an LTI system - h(t) = 0 for t < 0; mandatory for real-time hardware.
Total response - zero-input plus zero-state; equivalently natural plus forced.
FIR - always stable, no feedback; IIR - recursive, compact, stability must be checked.
Correlation - similarity/delay measure; R_xx(0) equals signal energy and is the maximum of R_xx(τ).
Even-odd decomposition - every signal is uniquely a sum of an even and an odd part.

Formulas and examples

Energy E = ∫_{−∞}^{∞} |x(t)|² dt (joules, for a voltage across 1 Ω); E = Σ_{n=−∞}^{∞} |x[n]|²
Average power P = lim_{T→∞} (1/2T) ∫_{−T}^{T} |x(t)|² dt (watts); P = lim_{N→∞} (1/(2N+1)) Σ_{−N}^{N} |x[n]|²
Periodic signal power P = (1/T₀) ∫_{T₀} |x(t)|² dt; for x(t) = A cos(ω₀t + φ), P = A²/2
Period conditions: CT T₀ = 2π/ω₀; DT N = 2πm/ω₀ (smallest integer N, m integer)
Even/odd parts: xₑ(t) = ½[x(t) + x(−t)], xₒ(t) = ½[x(t) − x(−t)]; E = Eₑ + Eₒ
Sifting: ∫ x(t) δ(t − t₀) dt = x(t₀); δ(at) = δ(t)/|a|
Convolution: y(t) = ∫_{−∞}^{∞} x(τ) h(t − τ) dτ; y[n] = Σ_{k=−∞}^{∞} x[k] h[n − k]
Causal LTI: y(t) = ∫₀^t x(τ)h(t − τ)dτ; y[n] = Σ_{k=0}^{n} x[k]h[n − k]
Step response: s(t) = ∫₀^t h(τ)dτ, h(t) = ds/dt; s[n] = Σ_{k=0}^{n} h[k]
BIBO stability: ∫|h(t)|dt < ∞ (CT); Σ|h[n]| < ∞ (DT)
Autocorrelation: R_xx(τ) = ∫ x(t) x(t + τ) dt, R_xx(0) = E, R_xx(τ) = R_xx(−τ)
First-order CT system dy/dt + (1/τ)y = Kx/τ, step input: y(t) = K(1 − e^{−t/τ})u(t), time constant τ (s)

Symbols: x = input signal, y = output, h = impulse response, t in seconds, n a sample index, ω in rad/s, f in Hz, T₀ and N the fundamental periods, A amplitude (volts), τ the delay variable or the time constant.

Example 1 - classify by energy and power.

x₁(t) = 10 cos(50πt): periodic, T₀ = 2/50 = 0.04 s. P = 10²/2 = 50 W, E = ∞ ⇒ power signal.
x₂(t) = e^{−2t}u(t): E = ∫₀^∞ e^{−4t}dt = 1/4 = 0.25 J, P = 0 ⇒ energy signal.
x₃[n] = 3(0.8)ⁿu[n]: E = Σ₀^∞ 9(0.64)ⁿ = 9/(1 − 0.64) = 9/0.36 = 25 J ⇒ energy signal.
x₄(t) = t·u(t): E = ∞ and P = lim (1/2T)∫₀^T t²dt = lim T²/6 = ∞ ⇒ neither.

Example 2 - periodicity.

x(t) = cos(10πt) + sin(4πt): T₁ = 2π/10π = 0.2 s, T₂ = 2π/4π = 0.5 s. Ratio 0.2/0.5 = 2/5, rational ⇒ periodic. T₀ = LCM(0.2, 0.5) = 1 s (5 cycles of the first, 2 of the second).
x[n] = cos(3πn/8): N = 2πm/(3π/8) = 16m/3 ⇒ smallest m = 3 gives N = 16 samples.
x[n] = cos(n): ω₀/2π = 1/2π, irrational ⇒ not periodic.

Example 3 - CT convolution of two pulses. x(t) = u(t) − u(t−2) (unit height, width 2), h(t) = u(t) − u(t−3) (unit height, width 3).

Overlap grows from t = 0 to t = 2: y(t) = t.
Full overlap for 2 ≤ t ≤ 3: y(t) = 2 (the shorter width).
Overlap shrinks for 3 ≤ t ≤ 5: y(t) = 5 − t; y = 0 outside 0 to 5.
Result: a trapezoid of base 2 + 3 = 5 s, peak 2. Check by area: ∫y = ½·2·2 + 1·2 + ½·2·2 = 2 + 2 + 2 = 6 = (area of x)(area of h) = 2 × 3 = 6. ✓

Example 4 - DT convolution and stability. x[n] = {1, 2, 3} for n = 0,1,2; h[n] = {1, 1, 1} for n = 0,1,2.

y[0] = 1·1 = 1; y[1] = 1·1 + 2·1 = 3; y[2] = 1 + 2 + 3 = 6; y[3] = 2 + 3 = 5; y[4] = 3.
y[n] = {1, 3, 6, 5, 3}, length 3 + 3 − 1 = 5. Check: Σy = 18 = (1+2+3)(1+1+1) = 6 × 3 = 18. ✓
Since h is FIR, Σ|h[n]| = 3 < ∞ ⇒ stable. For h[n] = (0.5)ⁿu[n], Σ = 1/(1 − 0.5) = 2 ⇒ stable; for h[n] = 2ⁿu[n] the sum diverges ⇒ unstable; for h[n] = u[n] the sum diverges ⇒ unstable even though h is bounded.

Example 5 - property tests and step response.

System	Linear?	Time-invariant?	Causal?	Stable?
y[n] = 3x[n] − x[n−2]	Yes	Yes	Yes	Yes (Σ	h	= 4)
y[n] = n·x[n]	Yes	No (n multiplies)	Yes	No (gain grows with n)
y[n] = x[2n]	Yes	No (decimation)	No (uses x[2] at n = 1)	Yes
y(t) = x²(t)	No	Yes	Yes	Yes
y(t) = x(t) + 4	No (non-zero output for zero input)	Yes	Yes	Yes
y(t) = ∫₋∞^t x(τ)dτ	Yes	Yes	Yes	No (integrator)

CT step response: dy/dt + 5y = 2x with x = u(t), y(0⁻) = 0. Here h(t) = 2e^{−5t}u(t), τ = 1/5 = 0.2 s, and y(t) = (2/5)(1 − e^{−5t}) = 0.4(1 − e^{−5t}). At t = 0.2 s, y = 0.4(1 − 0.3679) = 0.2528; at t = 0.4 s, y = 0.4(1 − 0.1353) = 0.3459; steady state 0.4.
DT step response: y[n] − 0.5y[n−1] = x[n] ⇒ h[n] = (0.5)ⁿu[n], s[n] = 2(1 − 0.5^{n+1}): s[0] = 1, s[1] = 1.5, s[2] = 1.75, s[∞] = 2 (equals Σh). ✓

Where it is used

Communication and modem design: channel modelled as an LTI impulse response; equalisers are the inverse systems - the daily work of modem teams at Qualcomm India, MediaTek, Signalchip and Saankhya Labs.
Embedded DSP and audio/medical products: FIR and IIR filters implemented on TI, ADI, ARM Cortex-M and RISC-V cores for hearing aids, ECG machines (BPL, Allengers), power-line and motor-drive sensing.
Aerospace and defence electronics: ISRO telemetry and tracking chains, BEL and DRDO (LRDE) radar and sonar processing where matched filtering is literally correlation; SAMEER and BARC instrumentation.
Control and instrumentation roles: step and impulse response measurements, time constants, settling behaviour of plant transfer functions in L&T, Siemens, ABB and Honeywell India projects.
Image and video processing: 2-D convolution as the basic operation in cameras, ISPs and CNN layers - the same convolution sum extended to two indices.
Exams and selection tests: a standalone GATE EC/EE/IN section, and the prerequisite block in ISRO/DRDO/BARC/BSNL JTO/ESE question papers and in most campus aptitude-plus-technical screens for VLSI and DSP roles.

Interview questions

Q: What makes a system LTI, and why is that so useful? A: It obeys superposition and shift invariance; then the impulse response alone determines the output for any input through convolution, and frequency-domain analysis becomes valid.
Q: Is y(t) = x(t) + 5 a linear system? A: No. A linear system must give zero output for zero input, and superposition fails because the constant is not scaled. It is affine, not linear.
Q: Why is cos(n) not a periodic discrete-time signal? A: Periodicity needs ω₀ = 2πm/N with integers m and N, i.e. ω₀/2π rational; for ω₀ = 1 rad/sample the ratio 1/2π is irrational, so no integer period exists.
Q: State the BIBO stability condition and test h[n] = u[n]. A: Σ|h[n]| must be finite; for the unit step the sum diverges, so the accumulator is unstable although its impulse response is bounded.
Q: How do you get the impulse response if you can only measure the step response? A: Differentiate it in CT, h(t) = ds/dt; take the first difference in DT, h[n] = s[n] − s[n−1].
Q: Two LTI blocks are cascaded. Does order matter? A: No - convolution is commutative and associative, so the overall impulse response is h₁*h₂ either way (assuming ideal, loading-free blocks).
Q: How many samples does the convolution of a 10-point and a 25-point sequence have? A: 10 + 25 − 1 = 34 samples.
Q: What is the difference between zero-input and zero-state response? A: Zero-input response comes from initial conditions with no input; zero-state response comes from the input with zero initial conditions; the total response is their sum.
Q: Why must a real-time filter be causal, and what if the application is off-line? A: A real-time filter cannot use future samples; in off-line processing the whole record is stored, so non-causal (zero-phase) filtering is allowed.
Q: Is y[n] = x[2n] time-invariant? A: No. Downsampling makes the output depend on which input samples line up with even indices, so a shift of the input is not reproduced as an equal shift of the output.
Q: What does δ(2t) = δ(t)/2 mean physically? A: Compressing an impulse halves its area, and since the impulse is defined by unit area, the scaled impulse must be written with the 1/2 factor.
Q: How are convolution and correlation related? A: Correlation of x and y equals convolution of y with the time-reversed x; correlation measures similarity and delay, convolution gives a system's output.
Q: Can a system be BIBO stable but internally unstable? A: Yes - if an unstable internal mode is cancelled by a zero, the input-output behaviour looks bounded while the state grows; state-space eigenvalue analysis exposes it.

Common mistakes

Shifting and scaling in the wrong order while sketching x(at − b); always rewrite as x(a(t − b/a)) first.
Forgetting the time-reversal (folding) step in convolution, or not splitting the answer into the correct ranges of t where the overlap limits change.
Calling a system stable because h is bounded; the test is absolute integrability or summability (u(t) and u[n] fail it).
Treating a DT sinusoid like a CT one - assuming it is always periodic, or missing that frequencies differing by 2π are identical.
Computing average power by averaging the signal instead of its square, or using A² instead of A²/2 for a sinusoid.
Adding periods instead of taking the LCM for a sum of periodic signals, and declaring periodicity when the ratio of periods is irrational.
Testing only homogeneity (scaling) and skipping additivity when checking linearity, or ignoring that a non-zero output for zero input already rules it out.
Using an impulse response to predict the output of a system whose coefficients depend on time, or one that is non-linear - convolution applies only to LTI systems.
Mixing up natural/forced with zero-input/zero-state decomposition, and dropping initial-condition terms in difference-equation problems.