Signals & Systems MCQ – 26 previous year questions with solutions

h(t)=e^{-|t|} is nonzero for t<0, so the system is not causal. But ∫|h(t)|dt over all time is finite (=2), so the LTI system is BIBO stable. Therefore it is stable but not causal.

For an LTI system, output is the convolution of input and impulse response. Convolution is commutative, associative and distributive. Key relation: y=x*h; DT: y[n]=Σ x[k]h[n−k].

Exam focus: Flip, shift, multiply, sum. Common trap: Convolution usually increases finite-sequence length; it does not always reduce it.

The autocorrelation of a rectangular pulse is triangular. As one pulse slides across the other, their overlap length decreases linearly with shift; for an original pulse duration T, nonzero overlap exists from −T to +T, giving total triangular duration 2T.

Convolution is the bridge between an input signal and the impulse response of an LTI system. The time-reversal step arises because the kernel is h[n−k] or h(t−τ), not h[k−n]. Key relation: DT: y[n]=Σ_{k=−∞}^{∞}x[k]h[n−k]. CT: y(t)=∫x(τ)h(t−τ)dτ. Finite DT length: Ly=Lx+Lh−1.

Exam focus: Graphical convolution sequence: fold → shift → multiply → sum. Common trap: Confusing convolution with element-wise multiplication or assuming output length always decreases.

A damped sinusoid is the imaginary part of a complex exponential e^{(-a+jω)t}. For an LTI system, a complex exponential is an eigenfunction: the system changes only its complex gain, not the exponential's pole. Therefore the output retains the same decay rate a and angular frequency ω; only magnitude and phase change.

The impulse response uniquely characterizes an LTI system for a specified input-output relationship because any input can be decomposed into shifted impulses and the output follows by convolution. This concept applies in both continuous and discrete time. Key relation: y(t)=x(t)*h(t); y[n]=x[n]*h[n].

Exam focus: For LTI questions, think 'impulse response + convolution'. Common trap: Impulse response is not a discrete-time-only concept.

The impulse response uniquely characterizes an LTI system for a specified input-output relationship because any input can be decomposed into shifted impulses and the output follows by convolution. This concept applies in both continuous and discrete time. Key relation: y(t)=x(t)*h(t); y[n]=x[n]*h[n].

Exam focus: For LTI questions, think 'impulse response + convolution'. Common trap: Impulse response is not a discrete-time-only concept.

The sequence defined as 1 for n≥0 and 0 for n<0 is the discrete-time unit step u[n]. It switches from 0 to 1 at n=0 and remains at 1 thereafter.

Energy and power are mutually exclusive classifications for ordinary nonzero signals: an energy signal has finite total energy and zero average power, while a power signal has finite nonzero average power and infinite energy. A finite-duration bounded signal normally has finite energy. Key relation: E=∫|x(t)|²dt; P=lim(T→∞)(1/2T)∫_{−T}^T|x(t)|²dt.

Exam focus: Periodic nonzero → usually power signal, not energy signal. Common trap: Repetition does not make a periodic signal finite-energy.

The unit ramp r(t) differentiates to the unit step u(t), and differentiating once more gives the unit impulse δ(t). Therefore δ(t) = d²r(t)/dt².

A monotonic signal consistently moves in only one direction (non-increasing or non-decreasing) over its domain. Simplify each candidate and/or inspect its derivative; the keyed choice is the expression that reduces to a monotonic function.

A continuous-time signal is periodic only if there exists at least one finite positive T such that x(t+T)=x(t) for all t. If no positive finite T satisfies that condition, the signal is non-periodic (aperiodic).

A nonzero periodic signal repeats indefinitely; if it were zero for all positive time as a purely anticausal signal, periodic repetition would force it to be zero everywhere. An energy signal, however, need not have finite duration—for example an exponentially decaying signal can extend indefinitely yet have finite energy. Hence statement 1 is true and statement 2 false: option A.

The Laplace transform must be considered together with its region of convergence. For a rational LTI system, BIBO stability requires the ROC to include the jω axis; causality imposes a right-sided ROC for a causal rational system. Key relation: X(s)=∫x(t)e^(−st)dt; stability ⇒ jω axis lies in ROC.

Exam focus: Transform expression + ROC identifies the time signal. Common trap: Never place a pole inside the ROC.

A discrete-time signal is defined at discrete indices, commonly integer n, and discrete-time systems are often implemented by digital processors. Linear constant-coefficient difference equations are a standard way to describe such systems. Key relation: Example LTI form: y[n]+a1y[n−1]=b0x[n]+b1x[n−1].

Exam focus: Time discreteness and system linearity are separate properties. Common trap: Do not assume a digital/discrete system must be nonlinear.

A system is linear when input, output and output derivatives appear only to the first power and are not multiplied together; coefficients may still depend on time. In option B, dy/dt+2t y=t²x satisfies this condition. The other listed equations contain squared signal terms. Therefore option B is linear.

The term multiplying log(x) can become unbounded as x approaches 0⁺, so for the system to avoid that unbounded contribution its coefficient must be zero. Solve k²−3k−4=0, giving k=4 or k=−1.

Fourier convergence results determine the limiting value to which a Fourier series (an infinite trigonometric series) converges. Under the standard convergence conditions, the series converges to the function at continuity points and to the midpoint of a finite jump at discontinuities. In the wording of this MCQ, the applicable purpose is the limit of an infinite series.

For an ideal symmetric square wave, only odd harmonics are present and the nth harmonic amplitude is proportional to 1/n. Harmonic power is proportional to amplitude squared, so P7/P5=(5/7)²≈0.51, nearest to 0.5.

Fourier methods decompose signals into frequency components. Fourier series represents periodic signals as harmonically related sinusoids or equivalent complex exponentials; non-periodic signals are handled by Fourier transform rather than ordinary Fourier series. Key relation: Real even x(t) ⇒ X(ω) real and even.

Exam focus: Symmetry can eliminate sine/imaginary terms immediately. Common trap: Fourier series is fundamentally for periodic signals.

Periodic signals are naturally described by Fourier series, while non-periodic signals use Fourier transforms. For LTI systems, H(jω) directly shows how each sinusoidal frequency component is scaled and phase shifted. Key relation: y=x*h ⇔ Y(jω)=X(jω)H(jω).

Exam focus: Series for periodic, transform for non-periodic—both belong to frequency-domain analysis. Common trap: Thinking “frequency domain” means “periodic signals only”.

A real even time-domain signal has a real even Fourier transform and zero odd component. Fourier series represents periodic signals as harmonically related sinusoids or equivalent complex exponentials; non-periodic signals are handled by Fourier transform rather than ordinary Fourier series. Key relation: Real even x(t) ⇒ X(ω) real and even.

Exam focus: Symmetry can eliminate sine/imaginary terms immediately. Common trap: Fourier series is fundamentally for periodic signals.

The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. Anti-aliasing filters limit input bandwidth before sampling, and ideal reconstruction uses a low-pass interpolation filter. Key relation: f_s≥2f_max; Nyquist rate=2f_max.

Exam focus: Filter before the ADC, not after aliasing has already occurred. Common trap: Oversampling reduces aliasing risk; it does not create aliasing.

The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. Anti-aliasing filters limit input bandwidth before sampling, and ideal reconstruction uses a low-pass interpolation filter. Key relation: f_s≥2f_max; Nyquist rate=2f_max.

Exam focus: Filter before the ADC, not after aliasing has already occurred. Common trap: Oversampling reduces aliasing risk; it does not create aliasing.

The Laplace transform must be considered together with its region of convergence. The ROC excludes poles, and the same algebraic transform can represent different time signals with different ROCs. Key relation: X(s)=∫x(t)e^(−st)dt; stability ⇒ jω axis lies in ROC.

Exam focus: Transform expression + ROC identifies the time signal. Common trap: Never place a pole inside the ROC.

The Z-transform is the discrete-time counterpart of the Laplace transform. For an LTI discrete-time system, BIBO stability requires the unit circle to lie in the ROC. Key relation: X(z)=Σx[n]z^(−n); stability ⇒ |z|=1 lies in ROC.

Exam focus: ROC position tells causality/stability information. Common trap: Poles are excluded from the ROC.