EC202 SIGNALS & SYSTEMS PREVIOUS ANSWER KEY

1. MODIFIED -
APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
FOURTHSEMESTER B.TECH DEGREE EXAMINATION, APRIL 2018
Course Code: EC202
Course Name: SIGNALS & SYSTEMS
Scheme of Valuation
(Scheme of evaluation (marks in brackets) and answers of problems/key)
PART A
Answer any two full questions, each carries 15 marks. Marks
1 a) Condition for periodicity N, x[n+N]=x[n] ---- 1 Mark
Ans: N = 6 ---- 1 Mark
(2)
b) i) Proper Time shifting, Time reversal and Time scaling and respective plots:
f(t+3), f(-t+3) and f(3-2t) – 1 Mark each = 3 Marks
ii) Definition of energy of the signal. – 1 Mark
Energy computation and final answer E= 11/3 – 3 Marks
(7)
c) i) Nonlinear and Stable: Answer with proper arguments – 3 Marks
ii) Linear and Stable: Answer with proper arguments – 3 Marks
(6)
2 a) i) Plots of f(t) and g(t) with proper markings on the x and y axes – 1 Mark each
= 2 Marks
ii) Definition of convolution – 1 Mark
( ) ∗ ( ) =
0, < 0
2( − 1), 0 ≤ < 2
2( − ), 2 ≤ < 4
0, ≥ 4
(2)
(7)
b) For complex signals ( ) = ∗( ) ( − )
For Real signals ( ) = ( ) ( − )
Any one definition – 1 Mark
For complex signals ( ) = ∗( ) ∗ (− )
For Real signals ( ) = ( ) ∗ (− )
Any one relation – 1 Mark
(2)
c) h[n] must be absolute summable for the system to be stable – 1 Mark (4)

2. MODIFIED -
h[n] = u[n] is unstable with proper argument – 1 Mark
h[n] must be a causal signal for the system to be causal – 1 Mark
System is causal – 1 Mark
3 a) x[n]*x[n] = {1, -2, 3, -4, 3, -2, 1} using any appropriate method – 6 Marks
Since the time origin is misaligned in the question paper, need not check for the
time origin in the answer.
(6)
b) Time invariance – Concept description – 1 Mark
Checking for time invariance of the system – Time invariant - 2 Marks
(3)
c) (i) Computation of Energy E = infinity – 1 Mark
Computation of Power P= A2
/2, Finite power- so power signal – 2 Marks
(ii) Computation of Energy E = infinity – 1 Mark
Computation of Power P= 1/2, Finite power- so power signal – 2 Marks
(6)
PART B
Answer any two full questions, each carries 15marks.
4 a) CTFS – Analysis and synthesis equations – 1 Mark each – 2 Marks
=
( )
+
( )
– Computation – 6 Marks
=
1
4
, = 1
0, = , ≠ 1
1
1− 2 , =
- 2 Marks
Another approach: Treating the waveform as product of sinusoid and pulse
train and finding Fk as the convolution of the corresponding CTFS.
For sinusoid = , = , For pulse =
/
( )
= Σ = + =
1
4
(
− 1
2
+
+ 1
2
)
Form of the final answer may differ based on the approach. Any correct
approach may be given credit.
(10)
b) Statement of Parseval’s theorem for CTFT – 2 Marks
Proof – 3 Marks
(5)
5 a) (i) fm = 150 Hz – 1 Mark, fN = 2fm = 300 Hz. – 1 Marks (2)

3. MODIFIED -
(ii) - 3 Marks
- 3 Marks
(iii) Original signal can be recovered when Fs = 400 Hz, but not with Fs = 200
Hz --1 Mark
(6)
(1)
b) ( ) = + , : ℜ( ) > -- X(s) with ROC -- 2 Marks
( ) = ( ) ( ), : ℜ( ) >
−1
3
; ( ) =
−5
+ 1
+
2
+
+
3
+
Partial fraction expansion - 3 Marks
( ) = −5 + 2 + 3 ( )– 1 Mark
(6)
6 a) i) ( ) = – 3 Marks, ROC: -a < Re(s) < a – 1 Mark
(ii)Let ( ) = sin( 0
+ ) ( ) = (sin 0 + cos 0 sin ) ( ) , ( ) =
0
( )2+ 0
2 +
( )
( )2+ 0
2 ; ( ) = − ( ); ( ) = 0
( + )2+ 0
2 +
( + )
( + )2+ 0
2 − ,
ROC: Re(s) > -a – 1 Mark (Note : use frequency shift property )
(9)
b) Equation for Fourier Transform of g(t) - 1 Mark
Calculation of ( ) with a>0 and a<0 – 2.5 Marks each – 5 Marks
( ) =
1
| |
( ⁄ )
(6)
PART C
Answer any two full questions, each carries 20 marks.
7 a) i) ( ) = ---2 Marks, ROC |z|>2 ----1 Mark
ii) ( ) = 1 ----1 Mark, ROC is the entire z plane – 1 Mark
(5)
b) i) x[n] is right-sided: ROC is outside the outermost pole |z|>2 --- 2 Marks
ii) DTFT converges: ROC includes the unit circle 2/3 < |z| < 2 ---- 2 Marks
ii) x[n] is left-sided: ROC is inside the innermost pole |z| < 2/3 ---- 2 Marks
(6)

4. MODIFIED -
c) Period of , N1 = 3 --- 1 Mark
Period of , N2 = 7 --- 1 Mark
Period of x[n] = LCM (N1, N2) = 21, Fundamental frequency = --- 1
Mark
DTFS synthesis equation --- 1 Mark
Finding Xk by comparison: =
, = +7, −7
, = +3
, = −3
0, ℎ
---- 1 Mark
Magnitude Spectrum: 2 Marks
Phase spectrum: 2 Marks
(9)
8 a)
i) ( ) = --- 2 Marks
ii) Sketch of ROCs: |z| <1/4 and |z|>1/4 --- 1 Mark each ---- 2 Marks
iii) h[n] is left sided => ROC is |z|<1/4 --- 1 Mark
ℎ = − − − 1 − − --- 3 Marks
(2)
(2)
(4)
b) i) ℎ = (0.5) = = 1 1 − 0.5⁄ ---- 4 Marks
ii) = = + = = { − + + } ,
− ≤ < ---- 3 Marks
Computation of = . ---- 1 Mark
= √
− ∅ ℎ ∅ = 0.5 ---- 4 Marks
(4)
(8)

5. MODIFIED -
9 a) Any 4 properties of Z transform – 4 Marks
Statement of convolution theorem including ROCs --- 2 Marks
Proof --- 4 Marks
(10)
c) Application of time shifting property --- 2 Marks
Application of frequency shifting property ---- 2 Marks
(4)
d) DTFT definition --- 1 Mark
=
( )
--- 5 Marks
(6)
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