Signal Representation, LTI Systems, Fourier, Laplace and Z Transforms

Section 4 of the GATE Electrical Engineering (EE) paper: the mathematics that turns a physical waveform — a voltage, a current, a torque — into an object that can be manipulated algebraically. Start with how continuous and discrete signals are represented and how shifting and scaling change them; then linear time-invariant (LTI) and causal systems, the class every practical circuit and control loop belongs to; then Fourier series and the sampling theorem that connects continuous signals to their discrete samples; then the Fourier, Laplace and Z transforms, three windows onto the same underlying signal; and finally RMS and average value, the two numbers that describe any periodic waveform's practical effect regardless of its exact shape.

1. Signal representation, shifting/scaling, and LTI/causal systems

A continuous-time signal x(t) is defined for every real t; a discrete-time signal x[n] is defined only at integer n, typically obtained by sampling a continuous signal. Time-shifting x(t - t0) delays the signal by t0 (moves it right if t0 > 0); time-scaling x(at) compresses it in time for |a| > 1 and stretches it for |a| < 1, and reverses it if a is negative. These operations commute in a specific order — shift then scale is not the same signal as scale then shift by the same numbers — which is exactly the standard slip on this topic.

A system is linear if it obeys superposition (response to ax1 + bx2 is ay1 + by2), time-invariant if a shift in the input produces exactly the same shift in the output with no other change, and causal if its output at any time depends only on present and past inputs, never future ones. LTI systems are the central object of this whole section because they are exactly the class for which convolution, the transfer function and frequency response all apply cleanly.

⚠️ Shift-then-scale is not scale-then-shift
To plot x(2t - 4), the safe method is to first replace t with the whole expression at once: this is x(t) evaluated at 2t - 4, which is a scale-by-2 of x(t - 2) — shift x(t) right by 2 first, THEN compress by 2 — not a shift-by-4 of x(2t). Working the two steps in the wrong order is the single most common error in this topic, and the fix is always to factor the argument as a*(t - t0) before applying either operation.

2. Fourier series and the sampling theorem

Any periodic signal (continuous or discrete) satisfying mild conditions can be written as a Fourier series — a sum of a DC term plus harmonically related sinusoids (or complex exponentials) at integer multiples of the fundamental frequency. A signal with half-wave symmetry contains only odd harmonics; one that is even (symmetric about t=0) has a cosine-only series, and one that is odd has a sine-only series — reading the symmetry off the waveform first, before computing a single coefficient, is the fast way to this class of question.

The sampling theorem states that a band-limited continuous signal, with no frequency content above fm, can be reconstructed exactly from samples taken at a rate fs greater than 2*fm (the Nyquist rate). Sampling below this rate causes aliasing: higher frequencies fold back and masquerade as lower ones in the sampled signal, and once aliased, that information is permanently lost — no amount of processing after sampling can undo it, which is why anti-aliasing filters are placed BEFORE the sampler, never after.

🎯 Why exactly 2*fm and not fm
A real signal's spectrum is symmetric about zero frequency (positive and negative frequency content mirror each other), so a band-limited signal actually occupies the range from -fm to +fm — a total span of 2fm. Sampling at fs creates a copy of this spectrum centred at every multiple of fs, and the copies stop overlapping only once fs exceeds the FULL span 2fm, not the one-sided fm.

3. Fourier Transform, Laplace Transform and Z Transform

Three transforms, three domains
TransformApplies toVariable
Fourier Transform (FT)continuous, aperiodic (and periodic, via impulses)jw (pure frequency)
Laplace Transformcontinuous, includes growing/decaying and unstable signalss = sigma + jw
Z Transformdiscrete-time signalsz (complex, discrete analogue of s)

The Fourier Transform exists only for signals that are absolutely integrable (or handled as a limiting case with impulses for periodic ones), which excludes a growing exponential; the Laplace Transform generalises it by adding a real, decaying exponential factor e^(-sigma*t), which can force convergence for signals the Fourier Transform cannot handle at all — this is why Laplace, not Fourier, is the standard tool for transient and stability analysis of circuits and control systems. Setting s = jw (sigma = 0) in a Laplace Transform recovers the Fourier Transform exactly where both exist, but only within the Laplace transform's own region of convergence (ROC).

🧠 The ROC decides which signal a Laplace or Z transform belongs to
The same algebraic expression for X(s) or X(z) can correspond to a right-sided, a left-sided, or a two-sided time-domain signal, depending on which Region Of Convergence is paired with it — the algebraic transform alone is ambiguous. A right-sided (causal) signal's ROC is a region to the RIGHT of its rightmost pole (in s) or OUTSIDE its outermost pole (in z); a left-sided (anticausal) signal's ROC lies on the other side. A stable causal LTI system's ROC must include the jw-axis (s-plane) or the unit circle (z-plane).

4. RMS and average value of periodic waveforms

The RMS (root-mean-square) value of a periodic waveform is the DC value that would dissipate the same average power in a resistor: Vrms = sqrt[(1/T) * integral of v(t)^2 dt over one period]. The average (mean) value over one full cycle of a symmetric AC waveform (sine, square) is zero, which is why average value for such waveforms is instead quoted over a HALF cycle when the number is meant to be useful — a pure sine wave's half-cycle average is (2/pi) times its peak, about 0.637 Vm.

RMS and average value for standard waveforms (peak Vm)
WaveformRMS valueAverage value (half cycle)Form factor (RMS/average)
Pure sine waveVm/sqrt(2) ~ 0.707 Vm(2/pi) Vm ~ 0.637 Vm~1.11
Square waveVmVm1.0
Triangular waveVm/sqrt(3) ~ 0.577 VmVm/2~1.155
⚠️ Form factor and peak factor are different ratios
Form factor = RMS value / average value (over the relevant half-cycle), and peak factor (crest factor) = peak value / RMS value — two different ratios that both equal exactly 1 for a square wave (since its RMS, average and peak are all Vm), which is the easy way to remember they are not the same quantity and neither is universally larger than the other across all waveform shapes.

Key takeaways

  • To transform x(2t - 4), factor the argument as 2*(t - 2) first: shift by 2, then scale by 2 — working the steps in the wrong order gives a different, wrong signal.
  • Anti-aliasing filters must sit BEFORE the sampler, because aliasing destroys information the moment sampling falls below the Nyquist rate (fs > 2*fm) and nothing downstream can recover it.
  • Laplace generalises Fourier by adding a decaying exponential e^(-sigma t), letting it handle growing and unstable signals the Fourier Transform cannot; setting s = jw recovers Fourier only within the Laplace ROC.
  • The same X(s) or X(z) expression can belong to different time-domain signals; the ROC (which side of the poles) is what disambiguates causal from anticausal.
  • A symmetric AC waveform's full-cycle average is zero by cancellation; average value is meaningfully quoted over a half cycle, and form factor (RMS/average) is not the same ratio as peak factor (peak/RMS).

Practice questions (9)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. A system's output at time t depends only on the input at times up to and including t. This system is described as

    1. time-invariant
    2. causal
    3. linear
    4. stable
    Show answer

    Answer: B — causal

    Depending only on present and past inputs, never future ones, is exactly the definition of causality — a property every physically realisable real-time system must have, independent of linearity, time-invariance or stability.

  2. To sketch x(2t - 4) from x(t), the correct sequence of operations is

    1. shift x(t) right by 4, then compress by 2
    2. shift x(t) right by 2, then compress by 2
    3. compress x(t) by 2, then shift right by 4
    4. compress x(t) by 2, then shift right by 2
    Show answer

    Answer: B — shift x(t) right by 2, then compress by 2

    Factoring the argument gives 2*(t - 2), so the correct reading is x(t) shifted right by 2 FIRST and then compressed by 2 — options C and D compress first with the wrong resulting shift amount, and option A shifts by the un-factored 4 instead of the correct 2.

  3. A periodic waveform with half-wave symmetry contains, in its Fourier series,

    1. only even harmonics
    2. only odd harmonics
    3. only a DC term
    4. every harmonic equally
    Show answer

    Answer: B — only odd harmonics

    Half-wave symmetry — the second half of the cycle is the negative mirror of the first — cancels every even harmonic and the DC term exactly, leaving only odd harmonics (fundamental, 3rd, 5th, ...) in the series, which is a shortcut worth checking before computing any coefficient.

  4. A signal is band-limited to 5 kHz. To reconstruct it exactly from its samples, the minimum sampling rate must be strictly greater than

    1. 5 kHz
    2. 10 kHz
    3. 2.5 kHz
    4. 20 kHz
    Show answer

    Answer: B — 10 kHz

    The Nyquist rate is 2fm = 25 = 10 kHz, and sampling must exceed this rate, not merely equal fm or half of it — 5 kHz (option A) is only the one-sided bandwidth, and using it instead of double it is the standard mistake.

  5. The Laplace Transform generalises the Fourier Transform chiefly by

    1. restricting attention to periodic signals only
    2. adding a real, decaying exponential factor that can force convergence for growing or unstable signals
    3. replacing continuous time with discrete time
    4. removing the need for a region of convergence
    Show answer

    Answer: B — adding a real, decaying exponential factor that can force convergence for growing or unstable signals

    Laplace multiplies the signal by e^(-sigma*t) before integrating, and choosing sigma large enough can make an otherwise divergent (growing or unstable) signal's transform converge — exactly what lets Laplace, unlike Fourier, handle transient and unstable-system analysis directly, and it still needs its own ROC.

  6. Which of the following are correctly true of the region of convergence (ROC)?

    1. A right-sided causal signal's ROC lies to the right of its rightmost pole (s-plane).
    2. The same X(s) expression can correspond to different time-domain signals depending on the ROC chosen.
    3. A stable causal LTI system's ROC must include the jw-axis.
    4. The ROC of a signal always includes every pole of its transform.
    Show answer

    Answer: A — A right-sided causal signal's ROC lies to the right of its rightmost pole (s-plane).; B — The same X(s) expression can correspond to different time-domain signals depending on the ROC chosen.; C — A stable causal LTI system's ROC must include the jw-axis.

    The first three are exactly the standard ROC rules. The ROC never includes a pole — by definition a pole is where the transform is infinite/undefined, so it is always excluded from (typically forms the boundary of) the ROC, which is why option D is false.

  7. The RMS value of a pure sinusoid with peak Vm is

    1. Vm
    2. Vm / sqrt(2)
    3. (2/pi) * Vm
    4. Vm / 2
    Show answer

    Answer: B — Vm / sqrt(2)

    Integrating sin^2 over a full cycle and taking the square root gives Vm/sqrt(2) (about 0.707 Vm) for RMS — the (2/pi) factor is instead the sine wave's HALF-CYCLE AVERAGE value, not its RMS, which is the usual point of confusion.

  8. Form factor of a periodic waveform is defined as

    1. peak value divided by RMS value
    2. RMS value divided by average value
    3. average value divided by peak value
    4. RMS value divided by peak value
    Show answer

    Answer: B — RMS value divided by average value

    Form factor = RMS/average, which for a sine wave is about 0.707/0.637 ≈ 1.11. Peak divided by RMS is a different ratio, the peak (crest) factor, and confusing the two names is the standard trap.

  9. A square wave alternates between +Vm and -Vm. Its form factor (RMS value divided by average value over a half cycle) is ___

    Numerical answer — type the value.

    Show answer

    Answer: 1

    A square wave's magnitude is constant at Vm throughout each half cycle, so both its RMS value and its half-cycle average value equal Vm exactly, making the form factor Vm/Vm = 1 — the smallest form factor any waveform can have.