GATE Electrical Engineering Signals and Systems Syllabus

Signals and Systems syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Signals and Systems in GATE Electrical Engineering, not a summary of it.

1. 1

   Representation of Signals

   2 topics

   • Continuous Time Signals
     • Shifting and Scaling Properties
   • Discrete Time Signals
     • Shifting and Scaling Properties
2. 2

   Systems

   1 topic

   • Linear Time Invariant Systems
     • Causal Systems
3. 3

   Fourier Series

   2 topics

   • Continuous Time Periodic Signals
   • Discrete Time Periodic Signals
4. 4

   Sampling Theorem

   overview

   Examined as a single unit within Signals and Systems — no further topic split in the official outline.

5. 5

   Applications of Fourier Transform

   2 topics

   • Continuous Time Signals
   • Discrete Time Signals
6. 6

   Laplace Transform

   overview

   Examined as a single unit within Signals and Systems — no further topic split in the official outline.

7. 7

   Z Transform

   overview

   Examined as a single unit within Signals and Systems — no further topic split in the official outline.

8. 8

   R.M.S. Value

   overview

   Examined as a single unit within Signals and Systems — no further topic split in the official outline.

9. 9

   Average Value Calculation

   1 topic

   • For Any General Periodic Waveform

Signals and Systems flashcards for GATE Electrical Engineering

23 of 50 cards from the Signals and Systems deck — real questions with worked answers.

1. What is a continuous-time (CT) signal?

   A signal $x(t)$ defined for every value of time $t$ over a continuous interval (the independent variable $t$ is real and takes a continuum of values).

2. What is a discrete-time (DT) signal?

   A signal $x[n]$ defined only at discrete instants, i.e. for integer values of $n$. It is often obtained by sampling a CT signal as $x[n]=x(nT_s)$.

3. Define the continuous-time unit impulse (Dirac delta) by its sifting property.

   $\delta(t)$ satisfies $\int_{-\infty}^{\infty} x(t)\,\delta(t-t_0)\,dt = x(t_0)$, with $\int_{-\infty}^{\infty}\delta(t)\,dt = 1$ and $\delta(t)=0$ for $t\neq 0$.

4. Define the discrete-time unit impulse $\delta[n]$.

   $\delta[n]=1$ for $n=0$ and $\delta[n]=0$ otherwise. Its sifting property is $\sum_{k=-\infty}^{\infty} x[k]\,\delta[n-k]=x[n]$.

5. What is the relationship between the CT unit step $u(t)$ and the unit impulse $\delta(t)$?

   $u(t)=\int_{-\infty}^{t}\delta(\tau)\,d\tau$ and conversely $\delta(t)=\dfrac{d\,u(t)}{dt}$.

6. What is the relationship between the DT unit step $u[n]$ and the unit impulse $\delta[n]$?

   $u[n]=\sum_{k=-\infty}^{n}\delta[k]$ and conversely $\delta[n]=u[n]-u[n-1]$ (the first difference).

7. For a CT signal $x(t)$, what does the time shift $x(t-t_0)$ represent for $t_0>0$?

   A delay (shift to the right) by $t_0$; the entire waveform is moved later in time. For $t_0<0$ it is an advance (shift left).

8. For the time-scaled CT signal $x(at)$, when does it compress and when does it expand?

   For $|a|>1$ the signal is compressed (sped up) by factor $a$; for $0<|a|<1$ it is expanded (stretched/slowed). If $a<0$ there is also time reversal.

9. How do you sketch the transformed signal $x(at+b)$ from $x(t)$ — what is the correct order of operations?

   Shift first, then scale: rewrite as $x(a(t+b/a))$. Equivalently, apply time shift by $b$ on $x(t)$ to get $x(t+b)$, then time-scale by $a$. (Doing scaling before shifting requires shifting by $b/a$.)

10. For DT signals, why is time scaling $x[an]$ fundamentally different from CT scaling?

    $n$ must remain an integer, so true compression/expansion isn't continuous. $x[2n]$ is decimation (downsampling, drops samples) and is generally non-invertible; expansion requires upsampling with inserted zeros.

11. What does the DT time shift $x[n-n_0]$ do for integer $n_0>0$?

    It delays the sequence by $n_0$ samples (shifts right). For $n_0<0$ it advances (shifts left). $n_0$ must be an integer.

12. Give the precedence rule for the DT operation $x[an+b]$.

    Shift the sequence by $b$ first to obtain $x[n+b]$, then time-scale (decimate/reverse) by $a$. Both $a$ and $b$ must be integers for the result to remain a valid DT signal.

13. Define a Linear Time-Invariant (LTI) system.

    A system that is both linear (satisfies superposition: additivity and homogeneity) and time-invariant (a time shift of the input produces an identical time shift of the output).

14. State the superposition (linearity) condition for a system with operator $T\{\cdot\}$.

    If $y_1=T\{x_1\}$ and $y_2=T\{x_2\}$, then $T\{a\,x_1+b\,x_2\}=a\,y_1+b\,y_2$ for all constants $a,b$.

15. State the time-invariance condition for a system.

    If input $x(t)$ gives output $y(t)$, then for every $t_0$ the shifted input $x(t-t_0)$ gives output $y(t-t_0)$ (the system's behavior does not change with time).

16. What completely characterizes an LTI system, and how is the output computed?

    The impulse response $h(t)$ (or $h[n]$). The output is the convolution: $y(t)=x(t)*h(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\,d\tau$, and $y[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]$.

17. Write the CT and DT convolution sum/integral definitions.

    CT: $y(t)=\int_{-\infty}^{\infty}x(\tau)\,h(t-\tau)\,d\tau$. DT: $y[n]=\sum_{k=-\infty}^{\infty}x[k]\,h[n-k]$.

18. List the three algebraic properties of convolution for LTI systems.

    Commutative: $x*h=h*x$. Associative: $(x*h_1)*h_2=x*(h_1*h_2)$. Distributive: $x*(h_1+h_2)=x*h_1+x*h_2$ (parallel systems add).

19. What is a causal system?

    A system whose output at any time depends only on present and past inputs, never future inputs. $y(t_0)$ depends only on $x(t)$ for $t\le t_0$.

20. State the impulse-response condition for an LTI system to be causal.

    $h(t)=0$ for all $t<0$ (CT), and $h[n]=0$ for all $n<0$ (DT).

21. State the impulse-response condition for an LTI system to be BIBO stable.

    The impulse response must be absolutely integrable/summable: $\int_{-\infty}^{\infty}|h(t)|\,dt<\infty$ (CT) or $\sum_{n=-\infty}^{\infty}|h[n]|<\infty$ (DT).

22. Is the system $y(t)=x(t+2)$ causal? Why?

    No. The output at time $t$ depends on a future input value $x(t+2)$, which violates causality (it is non-causal / anticipatory).

23. Define a memoryless (static) system and give an LTI example.

    A system whose output at time $t$ depends only on the input at the same time $t$. For an LTI system this means $h(t)=K\delta(t)$, i.e. $y(t)=Kx(t)$.

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