Assignment-1
Q(1).Analyse the block diagram of figure(a) and develop the relation between
y[n]and x[n].
Figure(a)
Q(2). Let 𝑥
̅̅̅[n],
𝑥2
̅̅̅[n],
and ̅̅̅[n]
𝑥3 be three periodic sequences with fundamental
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periods, 𝑁1 , 𝑁2 and 𝑁3 respectively. Is a linear combination of these three
periodic sequences a periodic sequence? If it is, what is its fundamental
periods?
Q(3). Determine the periodic conjugate symmetric and periodic conjugate
antisymmetric parts of the
following sequences:
𝑛
{x[n]}={A𝛼 },-N≤ n≤ N, where A and 𝛼 are complex numbers.
Q(4). Show that an absolutely summable sequence has finite energy, but a finite
energy sequence may not be absolutely summable.
Q(5). Compute the energy of the length-N sequence
2𝜋𝑘𝑛
x[n]=cos(
), 0≤ 𝑛 ≤ 𝑁 − 1.
𝑁
Q(6). Determine the average power and the following sequences:
(a) 𝑥1 [n] = 𝜇[n]
(b) 𝑥3 [n] = 𝐴0 𝑒 𝑗𝑤0𝑛
Q(7). Determine the fundamental period of the following periodic sequence:
𝑥
̃(n)
= 5sin(1.2𝜋n+0.65 𝜋)+4sin(0.8 𝜋n)-cos(0.8 𝜋n)
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Q(8). The second derivative y[n] of a sequence x[n] at time instant n is usually
approximately by y[n] = x[n+1]-2x[n]+x[n-1]. if y[n] and x[n] denote the
output and input of a discrete-time system, is the system linear? Is it timeinvariant? Is it causal?
Q(9). An algorithm for the calculation of the square root of a number 𝛼
Y[n] = x[n]-𝑦 2 [n-1]+y[n-1].
(A)
Where x[n]=𝛼𝜇[n] with 0< 𝛼 < 1.If x[n] and y[n] are considered as the input
and output of a discrete-time system, is the system linear or nonlinear? Is it
time-invariant? As n ∞, show that y[n] √𝛼 .Note that y[-1] is a suitable
initial approximation to √𝛼.
Q(10).A periodic sequence 𝑥̃[n] with a period N is applied as an input to an LTI
discrete-time system characterized by an impulse response h[n] generating an
output y[n].Is y[n] a periodic sequence? If it is, what is its period?
MATLAB
1.Write a MATLAB program to generate the following sequences and plot them
using the function stem:
(a) unit sample sequence𝛿[n],
(b) unit step sequence 𝜇[n],
(c) ramp sequence n𝜇[n].
The input parameters specified by the user are the desired length L of the
sequence and the sampling frequency 𝐹𝑇 in Hz. using this program generate the
first 100 samples of each of the above sequences with a sampling rate of 20kHz.
2. Write a MATLAB program to generate a sinusoidal sequence
x[n]=A cos(𝑤0 𝑛 + ∅) and plot the sequence using the stem function. The input
data specified by the user are the desired length, amplitude A, the angular
frequency 𝑤0 and the phase ∅ where 0< 𝑤0 < 𝜋 and 0≤ 𝜙 ≤ 2𝜋.
3.Write a MATLAB program to compute the square root using the algorithm of
Eq.(A) in the above problem 9 and show that the output y[n] of this system for
an input x[n]= 𝛼𝜇[n] with y[-1]=1 converges to √𝛼 as n
∞. Plot the error as
a function of n for several different values of 𝛼.How would you compute the
square-root of a number 𝛼 with a value greater than one?