A system is described by y(t) = 3x(t) + 5. Which is true?

Neither linear nor time-invariant

Convolution and correlation of two signals give the same result when

one of the signals is an even function

Which statements about sampling and aliasing are correct?

An anti-aliasing filter must act before the analog-to-digital converter.

Aliased components cannot be removed by any later digital processing.

A signal with no energy above half the sampling rate can be recovered exactly.

An anti-aliasing filter must act before the analog-to-digital converter.

Aliased components cannot be removed by any later digital processing.

Increasing the converter's resolution in bits prevents aliasing.

A signal with no energy above half the sampling rate can be recovered exactly.

Two LTI blocks with impulse responses h1 and h2 are connected in cascade, and the same two are connected in the opposite order. The overall impulse responses are

identical, because convolution is commutative

different, because the first block loads the second

identical only if both blocks are stable

different unless both impulse responses are even

Signals, LTI Systems, Convolution and Sampling

Section 4 of the GATE Instrumentation Engineering (IN) paper. Every instrument in the rest of this paper is a system that takes a signal in and gives a signal out, and this section is the language for saying what such a system does. Start with the impulse: a system that is linear and time-invariant is completely described by what it does to one sharp tap, and everything else — the response to a step, a sine, a recorded waveform — follows from that one answer by convolution. The second half is what happens when a continuous signal is read at intervals, which is what every data-acquisition card in Section 7 is doing, and the rule that decides whether the samples still contain the signal or a lie about it.

1. The signals worth naming

A signal is periodic if it repeats exactly after some time T, and the smallest such T is its period. Everything else is aperiodic. The distinction earns its place because a periodic signal has a Fourier SERIES — a sum of discrete harmonics — while an aperiodic one needs a Fourier TRANSFORM, a continuum of frequencies. Two signals do the heavy lifting in this section. The unit step turns something on and stays on. The unit impulse is the idealised sharp tap: zero width, unit area, and infinite height, which is not a function in the ordinary sense but is exactly what is needed to probe a system.

⚠️ Period of a sum

The sum of two periodic signals is periodic only if their periods have a rational ratio, and then the period is the LCM of the two, not the larger of them. sin(2t) + sin(3t) has periods pi and 2pi/3, ratio 3/2, so the sum repeats after 2pi. sin(t) + sin(pi*t) never repeats at all.

The impulse has one property that matters more than its shape: the sifting property. Integrating any continuous signal against a shifted impulse picks out the value of that signal at the shift. That single fact is why a system's response to an impulse tells you its response to everything.

2. Linear, time-invariant, and why both are needed

Linearity means scaling the input scales the output and adding two inputs adds their outputs. Time-invariance means delaying the input only delays the output, with no change of shape. A system with both is an LTI system, and an LTI system is completely characterised by its impulse response h(t): once h is known, the output for any input is the convolution of the input with h. Neither property alone is enough. A squarer is time-invariant and not linear. An amplifier whose gain drifts with the hour is linear and not time-invariant, and its output shape depends on when you applied the input.

Reading h(t) for the two properties that decide usability
Property	Condition on h	What it means physically
Causal	h(t) = 0 for t < 0	No output before the input arrives
BIBO stable	h is absolutely integrable	Every bounded input gives a bounded output

🎯 Why absolute integrability, not just decay

An impulse response can tend to zero and still fail. h(t) = 1/(1+t) decays to nothing, yet its integral diverges, and a bounded input exists that drives the output without bound. Decay is about the tail; stability is about the total area under it.

3. Convolution and correlation: same integral, different question

Convolution flips one signal, slides it across the other, and reports the overlap at every shift. That flip is the whole difference from correlation, which slides without flipping. Convolution answers what a system DOES to a signal; correlation answers how much two signals RESEMBLE each other at a given lag, which is what a radar or an ultrasonic flowmeter in Section 9 is computing when it looks for an echo. For a symmetric signal the two coincide, which is why the distinction is easy to forget and easy to be caught on.

Convolution is commutative, associative and distributive — so cascaded LTI blocks may be reordered on paper, whatever the wiring says.
Convolving with an impulse returns the signal unchanged; convolving with a shifted impulse shifts it.
The length of a convolution of two finite sequences is the sum of their lengths minus one.

🧠 A length check that costs one second

Before computing any discrete convolution, write down the expected length: 4 samples convolved with 3 gives 6. If the arithmetic produces any other count, the error is in the bookkeeping rather than the multiplication.

4. Sampling, and the frequency that is a lie

Sampling a continuous signal every T seconds makes copies of its spectrum spaced at the sampling frequency. If the signal contains nothing above half the sampling frequency, the copies do not touch and the original can be recovered exactly. If it does, the copies overlap and the parts that fold back arrive disguised as lower frequencies. That is aliasing, and nothing downstream can undo it: once two frequencies have produced identical samples, no filter, no processing and no amount of resolution can tell them apart again.

The practical consequence is the anti-aliasing filter, and it must sit BEFORE the converter. A low-pass filter in the analog path removes what would have folded; the same filter applied after digitising removes a band of the spectrum but leaves the aliased components exactly where they landed, because they are now indistinguishable from real signal at those frequencies.

ℹ️ Where the folded frequency lands

A component at f sampled at fs appears at the distance from f to the nearest multiple of fs. Sampling a 30 Hz tone at 50 Hz puts it at 20 Hz, because 50 - 30 = 20. It does not appear at 30 Hz and it does not vanish.

Key takeaways

An LTI system is its impulse response — h decides the output for every possible input, by convolution.
Causality is h(t) = 0 before zero; BIBO stability is h absolutely integrable. Decaying to zero is not enough.
Convolution flips before sliding, correlation does not — and the two agree only when one signal is symmetric.
Sample above twice the highest frequency present, and filter before the converter, never after.
Aliasing is not noise and cannot be removed downstream; it is a wrong frequency that now looks real.

Practice questions (8)

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.

A system is described by y(t) = 3x(t) + 5. Which is true?
1. Linear and time-invariant
2. Time-invariant but not linear
3. Linear but not time-invariant
4. Neither linear nor time-invariant
Show answer
Answer: B — Time-invariant but not linear
Delaying the input delays the output unchanged, so it is time-invariant. It is not linear: zero input gives an output of 5, and a linear system must map zero to zero. The constant offset is the whole of the failure — calling this linear because the graph is a straight line is the usual slip, and a straight line through 5 is affine rather than linear.
The signal x(t) = sin(2t) + sin(3t) is periodic with fundamental period
1. 2*pi
2. pi
3. 2*pi/3
4. It is not periodic
Show answer
Answer: A — 2*pi
The two components have periods pi and 2pi/3. Their ratio is 3/2, which is rational, so the sum repeats at the least common multiple of the two, namely 2pi. Answering pi takes the larger of the two periods rather than the LCM, and at t = pi the second component has completed one and a half cycles rather than a whole number of them.
An LTI system has impulse response h(t) = e^(-2t) for t >= 0 and zero otherwise. The system is
1. causal and BIBO stable
2. causal but not BIBO stable
3. BIBO stable but not causal
4. neither causal nor BIBO stable
Show answer
Answer: A — causal and BIBO stable
h vanishes for negative time, so no output precedes its input and the system is causal. The integral of the magnitude of h from zero to infinity is one half, which is finite, so it is BIBO stable. The trap is treating exponential decay as sufficient on its own: decay is necessary but the test is finite total area, and 1/(1+t) decays while its integral diverges.
A discrete sequence of 7 samples is convolved with a sequence of 4 samples. How many samples does the result have?
Numerical answer — type the value.
Show answer
Answer: 10
The convolution of sequences of lengths M and N has length M + N - 1, so 7 + 4 - 1 = 10. The minus one is there because both sequences contribute a sample at the first overlap position; answering 11 forgets it, and answering 7 confuses convolution with filtering a signal in place, which is what a running average of fixed output length would do.
Convolution and correlation of two signals give the same result when
1. both signals are causal
2. one of the signals is an even function
3. both signals are periodic
4. the signals have equal energy
Show answer
Answer: B — one of the signals is an even function
The only difference between the two operations is that convolution reverses one signal before sliding it. Reversing an even function leaves it unchanged, so the two integrals coincide. Causality says nothing about symmetry and equal energy says nothing about shape, so neither forces the reversal to be harmless.
A 30 Hz sinusoid is sampled at 50 Hz with no anti-aliasing filter. At what frequency, in Hz, does it appear in the sampled data?
Numerical answer — type the value.
Show answer
Answer: 20
The sampling rate is below twice the signal frequency, so the component folds. It reappears at the distance from 30 Hz to the nearest multiple of the sampling rate, that is 50 - 30 = 20 Hz. Answering 30 assumes the sample rate is adequate when it is not, and answering 25 quotes the Nyquist frequency, which is the boundary rather than where this component lands.
Which statements about sampling and aliasing are correct?
1. An anti-aliasing filter must act before the analog-to-digital converter.
2. Aliased components cannot be removed by any later digital processing.
3. Increasing the converter's resolution in bits prevents aliasing.
4. A signal with no energy above half the sampling rate can be recovered exactly.
Show answer
Answer: A — An anti-aliasing filter must act before the analog-to-digital converter.; B — Aliased components cannot be removed by any later digital processing.; D — A signal with no energy above half the sampling rate can be recovered exactly.
Filtering must precede conversion, because after conversion the folded components sit at frequencies where real signal also lives and are indistinguishable from it; that is also why no later processing removes them. Recovery is exact when nothing exceeds half the sampling rate. Resolution is a separate axis entirely: more bits reduce quantisation error and do nothing whatever about which frequencies folded.
Two LTI blocks with impulse responses h1 and h2 are connected in cascade, and the same two are connected in the opposite order. The overall impulse responses are
1. identical, because convolution is commutative
2. different, because the first block loads the second
3. identical only if both blocks are stable
4. different unless both impulse responses are even
Show answer
Answer: A — identical, because convolution is commutative
The cascade of two LTI systems has impulse response h1 convolved with h2, and convolution is commutative, so the order on paper makes no difference to the mathematics. Stability and symmetry are irrelevant to the commutation. The loading answer is the physically alert one and is exactly why this is an idealisation: real cascaded stages do load one another, which is why instrumentation amplifiers are buffered, but that effect is outside the LTI block description and is modelled by changing h rather than by reordering it.