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Signals and Systems - Analysis of Discrete Time Signals - Solved sample Problems - Important Short Questions and Answers: Analysis of Discrete Time Signals

**1. What is the relation between Z
transform and fourier transform of discrete time signal.**

X(ω)=X(Z)|z=e^{jω}.This means Z
transform is same as fourier transform when

evaluated on unit circle.

**2. Define region of convergence
with respect to Z transform.**

Region of convergence (ROC) is the area in Z
plane where Z transform convergence

.In other word, it is possible to calculate
the X(z) in ROC.

**3. State the initial value theorem
of Z transforms.**

The initial value of the sequence is given as,
X(0)=lim_{z->1}X(z).

**4. What is meant by aliasing? **

When the high frequency interferes with low
frequency and appears as low

then the phenomenon is called aliasing.

**5. Define Nyquist rate and Nyquist
interval.**

When
the sampling rate becomes exactly equal to ‘2W’ samples/sec,for a give
bandwidth of W hertz, then it is called Nyquist rate .’

Nyquist
interval is the time interval between any two adjacent samples.

Nyquist
rate =2W hz&Nyquist interval=1/2W seconds.

**6. Define unilateral Z-Transform or
one sided Z-transform**

The
unilateral Z-Transform of signal x(t) is given as,

The
unilateral and bilateral Z-Transforms are same for causal signals.

**7. State the final value theorem
for z-transform.**

The
final value of a sequence is given as,

**8. Define DTFT pair.**

DTFT,

**9. State the sampling theorem.**

•
A bandwidth signal of finite energy, which has no frequency components higher
than W hertz, is completely described by specifying the values of the signal at
instants of time separated by 1/2W seconds.

•
A band limited signal of finite energy, which has no frequency components
higher than W hertz, may be completely recovered from the knowledge of its
samples taken at the rate of 2W samples per second.

**10. Define two sided Z transform.**

The
z- transform of the DT signal is given by,

Here
‘z’ is the complex variable. The z- transform pair is denoted by,

**11. State the convolution property
of z transform. **

The
convolution states that,

That
is the convolution of two sequences in time domain is equivalent to
multiplication of their z-transforms.

**12. State parseval‟s theorem.**

Consider
the complex valued sequences x(n) and y(n).

If
x(n)----_X(k)

y(n)----_Y(k)

then
x(n)y*(n)=1/N X(k)Y*(k)

**13. Find Z transform of
x(n)={1,2,3,4} x(n)= {1,2,3,4}**

X(z)=
x(n)z-n

=
1+2z-1+3z-2+4z-3.

=
1+2/z+3/z2+4/z3.

**14. What z transform of (n-m)?**

By
time shifting property

Z[A
(n-m)]=AZ-m sinZ[ (n)] =1

**15. Obtain the inverse z transform
of X(z)=1/z-a,|z|>|a|**

Given
X(z)=z-1/1-az-1

By
time shifting property

X(n)=an.u(n-1)

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