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 π₯
Μ
Μ
Μ
[n],
π₯2
Μ
Μ
Μ
[n],
and Μ
Μ
Μ
[n]
π₯3 be three periodic sequences with fundamental
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periods, π1 , π2 and π3 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:
π
{x[n]}={AπΌ },-N≤ n≤ N, where A and πΌ 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
2πππ
x[n]=cos(
), 0≤ π ≤ π − 1.
π
Q(6). Determine the average power and the following sequences:
(a) π₯1 [n] = π[n]
(b) π₯3 [n] = π΄0 π ππ€0π
Q(7). Determine the fundamental period of the following periodic sequence:
π₯
Μ(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 πΌ
Y[n] = x[n]-π¦ 2 [n-1]+y[n-1].
(A)
Where x[n]=πΌπ[n] with 0< πΌ < 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] √πΌ .Note that y[-1] is a suitable
initial approximation to √πΌ.
Q(10).A periodic sequence π₯Μ[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 sequenceπΏ[n],
(b) unit step sequence π[n],
(c) ramp sequence nπ[n].
The input parameters specified by the user are the desired length L of the
sequence and the sampling frequency πΉπ 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(π€0 π + ∅) and plot the sequence using the stem function. The input
data specified by the user are the desired length, amplitude A, the angular
frequency π€0 and the phase ∅ where 0< π€0 < π and 0≤ π ≤ 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]= πΌπ[n] with y[-1]=1 converges to √πΌ as n
∞. Plot the error as
a function of n for several different values of πΌ.How would you compute the
square-root of a number πΌ with a value greater than one?