Two marks questions and answers

1. EC8553 Discrete Time Signal Processing
1 M.Starwin, AP/ECE
EC8553 Discrte Time Signal Processing
Two Mark Questions with Answers
Unit I
1. Calculate the 4 point DFT of the sequence x(n) = {1, 0, -1, 0}.
Solution:
X(k) = ∑ ( ) , 0 ≤ k ≤ N-1
= ∑ ( ) , 0 ≤ k ≤ 3
= ( ) +x(1) + x(2) + x(3)
= 1x1 + 0 + (-1)[cos πk – j sin π k] + 0
X(k) = 1 – [cos πk – j sin π k ] , 0 ≤ k ≤ 3
k = 0, X(0) = 1 – [cos 0 – j sin 0 ] = 1-1 =0
k = 1, X(1) = 1 – [cos π – j sin π ] = 1- (-1) = 2
k = 2, X(2) = 1 – [cos 2π – j sin 2π ] = 1- 1 = 0
k = 3, X(3) = 1 – [cos 3π – j sin 3π ] = 1- (-1) = 2
X(k) = { 0, 2, 0, 2}
2. What is twiddle factor?
Twiddle factor is a multiplication factor (i.e) multiplied with the input to
produce the transformed output.
Twiddle factor
Where, N is the length of DFT (or) number of input sequence.
3. Compare Radix 2 DIT, DIF FFT algorithm.
S.No Decimation in Time Algorithm Decimation in Frequency Algorithm
1 Time domain approach Frequency domain approach
2 Input is bit reversal order Output is regular order
3 Input is regular order Output is bit reversal order
4 The twiddle factor is multiplied
before the N point DFT.
The twiddle factor is multiplied
after the N point DFT.
4. What is the relationship between fourier transform and DFT?
Definition of DTFT, X( ) = ∑ ( )
W.K.T, ω = 2 π f =
X( ) = ∑ ( ) , 0 ≤ k ≤ N-1
= X(k)
X(k) = X( ) / ω = , 0 ≤ k ≤ N-1
5. State and prove periodicity property of DFT.
Statement : If x(n) X(k)
then, x(n +N) = x(n) for all ‘n’
X(k + N) = X(k) for all ‘k’

2. EC8553 Discrete Time Signal Processing
2 M.Starwin, AP/ECE
Proof:
Definition of DFT, X(k) = ∑ ( ) , 0 ≤ k ≤ N-1
Sub, k = k +N
X(k +N) = ∑ ( )
( )
= ∑ ( )
= ∑ ( ) (1), = 1
= ∑ ( )
X(k +N) = X(k)
Hence Proved
6. Test the causuality and stability of y(n) = sin x(n)
Causality:
Solution:
y(n) = sin x(n)
n = 0, y(0) = sin x(0)
n = -1, y(-1) = sin x(-1)
n = 1, y(1) = sin x(1)
The output of the system depends present input only, then the system is causal system.
Stability:
Solution:
y(n) = sin x(n)
∑ ( ) < ∞
∑ ( )
= ∑ ( )
Since, sin θ value ranges between 1 and -1. Hence, the system is stable system.
7. Is h(n) = ( ) + ( )- ( ) is stable and causal ? Justify.
Causal:
Solution:
n = 0, h(0) ( ) + ( ) ( )
The output of the system depends present, past and future. Hence, the system is non
causal system.
Stable:
Solution:
∑ ( ) < ∞
∑ ( )

3. EC8553 Discrete Time Signal Processing
3 M.Starwin, AP/ECE
∑ ( ) ( ) ( )
= ( ) + ( ) ( ) ( ) + ( ) ( ) ( ) +
( ) ( ) ( ) + ( ) ( )+…
= + = +
Hence, the system is stable system.
8. Define DT system.
Discrete Time(DT) systems accept and process the discrete time signals and
produce the output in the form of discrete time signals.
9. How do you obtain a digital signal from DT signal?
The digital signal can be obtained by passing the discrete time signal through
quantizer and encoder.
10. What is the relation between DTFT and DFT?
Definition of DTFT, X( ) = ∑ ( )
W.K.T, ω = 2 π f =
X( ) = ∑ ( ) , 0 ≤ k ≤ N-1
= X(k)
X(k) = X( ) / ω = , 0 ≤ k ≤ N-1
11. Compute the DFT of the sequence x(n) = {1, -1, 1,-1}.
Solution:
X(k) = ∑ ( ) , 0 ≤ k ≤ N-1
= ∑ ( ) , 0 ≤ k ≤ 3
= ( ) +x(1) + x(2) + x(3)
= 1- 1[cos – j sin ] + [cos πk – j sin π k] – 1[cos – j sin ],
0 ≤ k ≤ 3
k = 0, X(0) = 1- 1[cos – j sin ] + [cos 0 – j sin 0] – 1[cos – j sin ]
= 1-(1) + 1(1) – 1(1) = 1-1+1-1= 0
k =1, X(1) = 1- 1[cos – j sin ] + [cos π – j sin π ] – 1[cos – j sin ]
= 1-1(0-j) + 1(-1-0) – [0-j(-1)] = 1+j-1-j = 0
k =2, X(2) = 1- 1[cos π – j sin ] + [cos 2π – j sin 2π ] – 1[cos – j sin ]
= 1-1(-1) + 1 –(-1) = 4
k =3, X(3) = 1- 1[cos – j sin ] + [cos 3π – j sin 3π ] – 1[cos – j sin ]
= 1-1[0-j(-1)] + 1(-1) – ( -j ) = 1- j-1+j = 0
X(k) = { 0, 0, 4, 0}

4. EC8553 Discrete Time Signal Processing
4 M.Starwin, AP/ECE
Unit II
1. What is known as warping effect?
For low frequencies, the relationship between Ω and ω are linear. However,
for high frequencies, the relationship between Ω and ω are non-linear and distortion is
introduced in the frequency scale of the digital filter to that of the analog filter. This is
known as the warping effect.
2. Why impulse invariant method is not preferred in the design of IIR filter other than
LPF?
In impulse invariant technique, there is many to one mapping. Hence, high
frequencies are mapped on low frequencies and are suitable only for low pass filter
because of low resonant frequencies. Therefore, it is not suitable for high pass filters
and band pass filters of high resonant frequencies.
3. What are the requirements for the digital filter to be stable and causal ?
 A digital filter is causal if its impulse response h(n) = 0 for n < 0.
 A digital filter is stable if its impulse response h(n) is absolutely summable,
(i.e) ∑ ( ) < ∞
4. Discuss the need for prewarping.
The warping effect can be eliminated by prewarping the analog filter. This can
be done by prewarping the analog filter using the formula
Ω =
5. List the different types of filters based on frequency response
The different types of filters based on frequency response are
 Low pass filters
 High pass filters
 Band pass filters
 Band stop filters
6. What are the properties of bilinear transformation?
 The bilinear transformation provides one to one mapping.
 The bilinear transformation transforms jΩ axis in the s-plane in to the unit
circle in the z - plane.
 It avoids aliasing of frequency components.
7. What are the methods used for digitizing the analog filter into a digital filter?
There are four methods used for digitizing the analog filter
 Impulse invariant transformation
 Bilinear Transformation
 Approximation Derivatives
 Matched Z Transform
8. What is meant by frequency warping or prewarping ?
The effect of the non – linear compression at high frequencies can be
compensated, when the desired magnitude response is constant over frequency. This
compression can be compensated by prewarping the analog filter using the formula

5. EC8553 Discrete Time Signal Processing
5 M.Starwin, AP/ECE
Ω =
9. Define pass band.
The range of frequencies are passed through the filter, then it is called as pass band.
10. Use the backward difference for the derivative to convert analog LPF with system
function H(s) =
Solution:
H(z) = H(s) / s
= / s
=
=
H(z) =
Assume, T = 1 secs
H(z) = = =
( )
=
Unit III
1. List the disadvantages of FIR filters.
The disadvantages of FIR filters are
 The implementation of narrow transition band filters is very costly.
 Memory requirement and execution time are very high.
 Powerful computational facilities required for the implementation.
2. List the desirable window characteristics.
The desirable window characteristics are
 The central lobe of the frequency response of the window should contain most
of the energy and should be narrow.
 The highest side lobe level of the frequency response should be small.
 The side lobes of the frequency response should decrease in energy rapidly as
ω tends to π.
3. Draw the direct form realization realization of FIR system.

6. EC8553 Discrete Time Signal Processing
6 M.Starwin, AP/ECE
4. Write the steps involved in FIR filter design.
Step(i) : Draw the frequency response and convert the analog frequency into digital.
Step(ii): Find the transition width and select the window.
Step(iii): Determine the order of the filter N.
Step(iv): Determine the desired frequency response (n).
Step(v): Determine the window function w(n).
Step(vi): Determine the actual filter response (or) unit sample response
h(n) = (n) x w(n)
Step(vii): Determine the transfer function
H(z) = ∑ ( )
Step(viii): Determine the frequency response
Sub z =
H( ) = ∑ ( )
Step (ix): Draw the magnitude response | H( )| and phase response ∠H( )
5. What is Gibbs phenomenon?
The truncation of fourier series is known to introduce ripples in the frequency
response characteristics H(ω) due to the non- uniform convergence of a fourier series
at a discontinuity. The oscillatory behaviour near the band edges of the filters called
the Gibbs phenomenon.
6. List out the advantages of FIR filters.
The advantages of FIR filters are
 FIR filters have exact linear phase.
 FIR filters are always stable.
 FIR filters can be realized both recursive and non-recursive structures.
 Filters with any arbitrary magnitude response can be tackled using FIR
sequence.
7. What do you understand by linear phase response?
For a linear phase filter, θ( ) = τω, - π ≤ ω ≤ π, the linear phase filter does not
alter the shape of the original signal. If the phase response of the filter is non linear,
the output signal may be distorted one. In many cases, a linear phase characteristics is
required throughout the pass band of the filter to preserve the shape of a given signal
with in the pass band.
8. Give the equation specifying Hamming window and Hanning window.
Hamming window:
wH(n) = {0.54 – 0.46 cos ( ) , 0 ≤ n ≤ N -1
0, Otherwise
Hanning window:
wH(n) = {0.5 – 0.5 cos ( ) , 0 ≤ n ≤ N -1
0, Otherwise

7. EC8553 Discrete Time Signal Processing
7 M.Starwin, AP/ECE
Unit IV
1. Distinguish between fixed point arithmetic and floating point arithmetic.
S.No Fixed point arithmetic Floating point arithmetic
1 Small Dynamic Range It covers large dynamic range
2 Overflow occurs in addition Overflow occurs in multiplication
3 It provides uniform resolution It provides fine resolution
4 Fast operation Slow operation
5 Round off errors occur only for
addition.
Round off errors occur both in
addition and multiplication.
6 Used in small computers. Used in large general purpose
computers.
2. What are the methods used to prevent overflow?
There are two methods used to prevent overflow
 Saturation Arithmetic
 Signal Scaling
3. What is meant by finite word length effects in digital system?
In digital systems, the filter coefficients are stored in finite word length
registers by quantization. ( truncation or rounding ) because of that some errors occur
in the digital filters. These errors are called finite word length errors or finite word
length effects.
4. What is meant by dead band of the filter?
The amplitude of the output during a limit cycle are confined to a range of
values called the dead band of the filter. This limit cycle occurs as a result of
quantization effects in multiplication. Dead band =
5. What are the two kinds of limit cycle behaviour in DSP?
The two kinds of limit cycle behaviour in DSP are
 Zero-input limit cycle oscillation
 Overflow limit cycle oscillation
6. List the representations for which truncation error is analysed.
 Fixed point number representation.
 Floating point number representation.
7. What are the different types of fixed point representation?
The different types of fixed point representations are
 Sign Magnitude representation
 1’s complement representation
 2’s complement representation
8. Name the three quantization error due to finite word length registers in digital filters.
The three quantization errors are
 Input quantization error
 Product quantization error
 Coefficient quantization error

8. EC8553 Discrete Time Signal Processing
8 M.Starwin, AP/ECE
9. What does the truncation of data result in ?
The truncation of data result in discarding the bits less significant than least
significant bit that is retained.
10. Why is rounding preferred over truncation in realizing a digital filter?
 The quantization error due to rounding is independent of the type arithmetic.
 The mean of rounding error is zero.
 The variance of the rounding error signal is low.