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Signals and Systems- Assignment 01 (Course Topics: Signal Representation, Signal Properties, Elementary Signal Transformations) Instructions: Use Separate Assignment Notebook for the Signals and Systems Course; Date of Submission of Notebook: 10-09-2015; Max. Marks: 100; Office Location: 2nd Floor Director Office Building, Samantapuri Campus; PART-I: Problems in Signal Representations 1. Write the mathematical expressions of the following CT and DT elementary functions and Sketch those elementary functions: unit impulse function, unit step function (or Heaviside unit function), ramp function, rectangular pulse function, signum (or sign) function, sinc function, sinusoidal function, exponentially increasing sinusoidal function, exponentially decreasing sinusoidal function, real-valued exponential decaying function with σ < 0, real-valued exponential function with σ = 0, and real-valued exponential rising function with σ > 0.

2. A continuous-time signal
 is shown in Fig. I-2. Sketch and label each of the following signals. (a)
1 − ;

(b)
[ −  − 1];



(c)
 − 

Figure I.2: A finite-duration two-sided CT signal

3. Represent a piecewise constant function
 as shown in Fig. I.3 in terms of step functions. Determine the derivative of function
 and sketch its derivative function.

Figure I.3: A piecewise constant function


4. Given the continuous-time signal specified by
 = 

1 − ||  0, ℎ

determine the resultant discrete-time sequence
[ ] obtained by uniform sampling of
 with a sampling interval Ts of (a) 0.25 s, (b) 0.5 s, and (c) 1.0 s. Sketch the discrete sequence x[n] for the sample index, −5 <  < 5. 5. Using the discrete-time signals x1[n] and x2[n] shown in Fig. I-5, represent each of the following signals by a graph (or graphical representation) and by a sequence of numbers (or sequence representation) (a) (b) (c)

! []

=
! [] +
[]; [] = 4
! [] ! ! [] =
! [] ×
[]

Figure I.5: A finite duration discrete time signals

6. Find and sketch the first derivatives of the following signals (a)
 =  −  − %, % > 0 (b)
 = [ −  − %], % > 0 1,  > 0  (c)
 = ' = ( −1,  < 0 7. Express the waveforms of Figure I.7(a) and (b) as a sum of unit step functions.

Figure I.7(a): Symmetric rectangular pulse;

Figure I.7(b): Symmetric triangular wave

PART-II: Problems in Periodic and Non-Periodic Signals 1. Determine if the following CT signals are periodic. If yes, compute the fundamental period T0 and fundamental frequency for the CT signals: i)

x1(t) = sin(-5πt/8 + π/2)

ii)

x2(t) = |sin(-5πt/8 + π/2)|

iii)

x3(t) = sin(6πt/7) + 2cos(3t/5)

iv)

x4(t) = exp(j(5t + π/4))

v)

x5(t) = exp(j3πt / 8) + exp(πt / 86)

vi)

x6(t) = 2cos(4πt / 5) * sin2 (16t / 3)

vii)

x7(t) = 1+ sin20t + cos(30t + π/3)

2. Determine if the following DT signals are periodic. If yes, compute the fundamental period N0 for the DT signal : i)

x1[k] = 5 x (-1)k

ii)

x2[k] = exp(j(7πk / 4)) + exp(j(3k / 4))

iii)

x3[k] = exp(j(7πk / 4)) + exp(j(3πk / 4))

iv)

x4[k] = sin(3πk / 8) + cos(63πk / 64)

v)

x5[k] = exp(j(7πk / 4)) + cos(4πk / 7 + π)

vi)

x6[k] = sin(3πk / 8)cos(63πk / 64)

3. Show that the complex exponential signal
 =  )*+ , is periodic and that its fundamental period is 2./01 .

4. Show that the complex exponential sequence
[] =  )2+ 3 is periodic only if 2./Ω1 is a rational number.

5. Consider the sinusoidal signal
 = cos15

(a) Find the value of sampling interval  such that
[] =
[ ] is a periodic sequence. (b) Find the fundamental period of
[] =
[ ] if  = 0.1. seconds.

6. Let
!  and
 be periodic signals with fundamental periods T1, and T2, respectively. Under what conditions is the sum
 =
!  +
 periodic, and what is the fundamental period of
 if it is periodic?

7. Let
! [] and
[] be periodic signals with fundamental periods N1, and N2, respectively. Under what conditions is the sum
[] =
! [] +
[] periodic, and what is the fundamental period of
[] if it is periodic?

PART-III: Problems in Energy and Power Signals 1. Determine if the following CT signals are energy or power signals or neither. Compute the energy and power of the signals in each case: i)

x1(t) = cos (πt)sin(3πt)

ii)

x2(t) = exp(-2t)

iii)

x3(t) = exp(-j2t)

iv)

x4(t) = exp(-2t)u(t)

v)

x5(t) = 

vi) vii)

cos3. , − 3 ≤  ≤ 3 0 , ;ℎ  , 0 ≤  ≤ 2 x6(t) =

viii) x[n] =cos( n) ix) x) xi) xii) xiii)

?@

?

x[k] =  ) A B C 
[] = []


[] = [] − [ − 1]
[] = ∑3FGHI [E]


[] = ∑I KGL [ − J]

PART-VI: Problems in Even and Odd Signals 1. Determine if the following CT signals are even, odd, or neither even nor odd. In the latter case, evaluate and sketch the even and odd components of the CT signals: i)

x1(t) = 2sin (2πt)[2+ cos(4πt)]

ii)

x2(t) = t2 + cos(3t)

iii)

x3(t) = exp(-3t)sin(3πt)

iv)

x4(t) = t sin(5t)

v)

x5(t) = t u(t)

vi) vii)

,, LM,M N, M,M> x6(t) = < H,BN, >M,MN L, OPOQROSO

x (t) =exp(jt)

2. Show that the product of two even signals or of two odd signals is an even signal and that the product of an even and an odd signal is an odd signal.

PART-V: Problems in impulse function 1. Evaluate the following expressions:

i)

TB ,B, A

ii)

WXY ,

iii)

*Z H!

UB, A B, V

,

*A H!

 − 2



0 − 2

2. Evaluate the following integrals i) ii) iii) iv) v) vi) vii) viii)

I

[HI − 1 − 2\ N

[HI − 1 − 2\ I

[N  − 1 − 2\ I

[HI2/3 − 53/4 − 5/6\ I

[HI exp  − 1sin . + 10/61 − \ I

[HI[sin c I

de >

f + exp −4 + 1]− − 1\

[HI[ut − 4 − ut − 8]sin 3πt /4 − 5\

!

[H !∑I KGHI  − 5J\

3. Describe the six properties of impulse function !

4. Verify the following equations: % = |k|  and − = 

PART-VI: Problems in Elementary Signal Transformations 1. A CT signal x(t) is shown in Fig. VI-1. Sketch and label each of the following signals. (a) x(t - 2);

(b) x(t + 2);

(c) x(2t);

(d) x(t/2);

(e) x (- t+2);

Figure VI.1: A finite duration CT signal x(t)

2. A DT signal x[n] is shown in Fig. VI-2. Sketch and label each of the following signals. (a) x[n–2];

(b) x[n+2];

(c) x[2n];

(d) x [n/2] ;

(e) x [-n + 2]

Figure VI-2: A finite duration DT signal x[n]

5. Consider the following signal: ,B ,H M,MH! ! ,H!M,M! x(t) =