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DEPARTMENT OF ECE

Sub. Code/Title: IT6502/ DIGITAL SIGNAL PROCESSING Sem: V

QUESTION & ANSWERS

UNIT I SIGNALS AND SYSTEMS

1. What are the steps involved in digital signal processing?

i) Converting the analog signal to digital signal. This is performed by A/D converter.

ii) Processing the digital signal by digital system.

iii) Converting the digital output signal from the digital system to analog signal using D/A

converter.

2. What are the advantages and applications of DSP?

Advantages:

i) The program can be modified easily for better performance.

ii) Better accuracy can be achieved by using adaptive algorithms.

iii) The digital signal can be easily stored and transported.

iv) The digital systems are cheaper than analog equivalent.

Applications:

i) Speech processing – Speech compression and decompression for voice storage system and for

Transmission and reception of voice signals.

ii) Communication – Elimination of noise by filtering and echo cancellation by adaptive filtering

in transmission channels.

3. What is meant by aliasing? How it can be avoided?

When the sampling frequency is less than twice of the highest frequency content of the signal, then

the aliasing is frequency domain takes place. In aliasing, the high frequencies of the signal mix with

lower frequencies and create distortion in frequency spectrum.

Aliasing can be avoided by two ways,

i) Sampling frequency must be higher than twice of highest frequency present in the signal.

ii) A low pass filter must be used before sampling to band limit the signal to some specific

frequency.

4. State the classification of discrete time signals.

The types of discrete time signals are

o Continuous-time and discrete time signals

o Even and odd signals

o Periodic signals and non-periodic signals

o Deterministic signal and Random signal

o Energy and Power signal

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5. Define impulse and unit step signal.

Impulse signal is defined as a signal having unit magnitude at n = 0 and zero for all other

values of n. It can be expressed as follows: Impulse signal, ∂ (n) = 1; n=0

Unit step signal is defined as a signal having unit magnitude for all values of n

Unit step signal, 𝑢(𝑛) = 1 ; 𝑛 ≥ 0

0; 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

6. What are energy and power signals?

The energy of a discrete time signal is defined as,

The average power of a discrete time signal is defined as,

7. What are even and odd signal?

A discrete time signal x (n) is called even or symmetric signal if it satisfies the condition

x(n) = x (-n)

A discrete time signal x (n) is called odd or anti-symmetric signal if it satisfies the condition

x(-n) = -x (n)

8. What is an LTI system?

A system is said to be linear time invariant (LTI), if it satisfies Superposition principle and time

invariance. For a linear system, if the response of the system to a weighted sum of the signals is

equal to the corresponding weighted sum of the response o f the system to each of the individual

input signals. For a time invariance system, if its input – output characteristics do not change with

time.

9. State the classification of systems.

• Static and dynamic system.

• Time invariant and time variant system.

• Causal and anti-causal system

• Linear and Non-linear system.

• Stable and Unstable system.

10. What is linear and nonlinear system?

A system is said to be linear, if satisfies the superposition principle. It states that if the response of

the system to a weighted sum of the signals is equal to the corresponding weighted sum of

the response of the system to each of the individual input signals.

H [a1 x1 (n) + a2 x2 (n))] = a1H[x1 (n)] + a2H[x2 (n)]

If the system does not satisfy the superposition principle then it is nonlinear system.

11. What is a shift invariant system? Give an example

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A system H is time invariant, if the response to a shifted version of the input is identical to a shifted

version of the response based on the un-shifted input.

Where represents a signal delay of k samples. Example: y (n) = x (n) - x (n-1)

12. Define static and dynamic systems (or) what is memory system and memory less system? Memory system: A system is said to be memory system if its output signal at any time depends on

the past values of the input signal. Ex. Circuit with inductors capacitors

Memory less system A system is said to be static or memory less, if its output at any instant n

depends at most on the input sample at the same time but not on past or future samples of the input.

Ex: An electronic circuit with resistors.

13. Define stable system (or) what is BIBO system?

A system is said to be Bounded Input –Bounded Output (BIBO) stable, if and only if

every bounded input produces a bounded output.

14. Define causality?

A system is said to be causal, if the output of the system at any time n depends only on

the present inputs, past inputs and past outputs but does not depends on future inputs and outputs. If

the system output at any time n depends on future inputs or outputs then the system is called non-

causal system.

15. What is BIBO stability? What is the condition to be satisfied for stability?

A system is said to be BIBO stable, if and only if every bounded input produces a

bounded output. The condition to be satisfied for the stability of an LTI system is that the impulse

response of the system should be absolutely sum able

16. Define sampling theorem

A continuous time signal can be completely represented in its samples and recovered back if the

sampling frequency Fs ≥ 2B, where ‘Fs ‘is the sampling frequency and ‘B ‘is the maximum

frequency present in the signal.

17. What is meant by region of convergence (ROC)?

The values of z for which the z transform can be evaluated (converged) are called the region of

convergence (ROC). The ROC must always be mentioned along with z-transform.

18. What are the different methods of evaluating inverse z-transform?

i) Partial fraction expansion

ii) Power series expansion

iii) Contour integration (Residue method)

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19. Check whether the system 𝒚(𝒏) = 𝒆𝒙(𝒏) is linear.

Answer: It is nonlinear

20. Check whether the system 𝒚(𝒏) = 𝒍𝒏[ 𝒙(𝒏)] is linear and time invariant.

Answer: It is nonlinear and time invariant

21. State the convolution property and time shifting property of z-transform.

Convolution property:

𝑥1(𝑛)𝑍

↔ 𝑋1(𝑧) and 𝑥2

(𝑛)𝑍

↔ 𝑋2(𝑧) then, 𝑥1(𝑛) ∗ 𝑥2(𝑛) 𝑋1(𝑧). 𝑋2(𝑧)

Time shifting property: 𝑥1(𝑛 − 𝑚)𝑍

↔ 𝑍−𝑚𝑋1(𝑧)

22. Determine which of the following signals are periodic and compute their fundamental period

x (t) is sinusoidal signal, which is periodic.

Fundamental period

That is rational. Hence the signal is periodic. The least common multiple of T1 and T2 is 2/5.

Hence the fundamental period,

23. What are the properties of convolution sum

The properties of convolution sum are

• Commutative property : x (n)*h (n) =h (n)*x (n)

• Associative law : x (n)*h1 (n)*h2 (n) =x (n) h1 (n)*h2 (n),

• Distributive law : x(n)*h1(n)+h2(n)=x(n)*h1(n)+x(n)*h2(n)

24. Find the Z- transform of 1,0,2,0,3

𝑋(𝑧) = 1 + 2𝑍−2+3𝑍−4 ROC: Entire Z- plane except Z=0

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25. What are the manipulations (or) operations performed over the sequence?

Amplitude modifications

o Amplitude scaling y[n] = a x[n], amplitude shift y[n] = x[n] +b

o Sum of two signals y[n] = x1[n] + x2[n]

o Product of two signals y[n] = x1[n] x2[n]

Time modifications

o Time shifting y[n] = x [n-k].

If ‘k’ is a positive integer it is called as delayed signal, otherwise it is called an

advanced signal

o Folding or reflection or time-reversal y[n] = x[ n]

o Time-scaling or down-sampling y[n] = x[2n]. (discard every other sample)

26. Check for linearity and stability of 𝒈(𝒏) = √𝒙(𝒏)

i) Since square toot is non linear, the system is nonlinear.

ii) As long as x (n) is bounded, its square root is bounded. Hence this system is stable.

27. State properties of ROC.

• The ROC does not contain any poles.

• When x (n) is of finite duration then ROC is entire Z-plane except Z=0 or Z=∞.

• If X (Z) is causal, then ROC includes Z=∞.

• If X (Z) is anti-casual, then ROC includes Z=0.

28. What is a continuous and discrete time signal?

Continuous time signal: A signal x(t) is said to be continuous if it is defined for all time t.

Continuous time signal arise naturally when a physical waveform such as acoustics wave

or light wave is converted into a electrical signal. This is affected by means of

transducer. (e.g.) microphone, photocell

Discrete time signal: A discrete time signal is defined only at discrete instants of time.

The independent variable has discrete values only, which are uniformly spaced. A

discrete time signal is often derived from the continuous time signal by sampling it at a

uniform rate.

29. What are deterministic and random signals?

Deterministic Signal: deterministic signal is a signal about which there is no certainty with respect

to its value at any time. Accordingly we find that deterministic signals may be modelled as

completely specified functions of time.

Random signal: random signal is a signal about which there is uncertainty before its actual

occurrence. Such signal may be viewed as group of signals with each signal in the ensemble having

different wave forms. (e.g.) The noise developed in a television or radio amplifier is an example for

random signal.

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30. Define cross correlation. Give their properties.

The cross correlation of signals x[n] and y[n] is

Properties of cross correlation:

o

o

o This is called the

Cauchy-Schwarz inequality.

o

31. What is auto correlation?

The autocorrelation of a signal is the cross correlation of the signal with itself:

32. What are elementary signals and name them?

The elementary signals serve as a building block for the construction of more complex signals.

They are also important in their own right, in that they may be used to model many physical signals

that occur in nature. There are five elementary signals. They are as follows

o Unit step function

o Unit impulse function

o Ramp function

o Exponential function

o Sinusoidal function

33. Define unit step, ramp and delta functions for Discrete Time.

Unit step function is defined as, U(n) = 1 ; for n ≥ 00 ; otherwise

Unit ramp function is defined as, r(n) = n ; for n ≥ 00 ; otherwise

Unit delta function is defined as δ(n) = 1 ; for n = 00 ; otherwise

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UNIT II FREQUENCY TRANSFORMATIONS

1. What is DFT pair? (or) Define DFT and IDFT (or) Define N-point DFT and N-point IDFT.

The sequence of N complex numbers x0... xN−1 is transformed into the sequence of N complex numbers

X0... XN−1 by the DFT according to the formula:

The N point IDFT (Inverse Discrete Fourier Transform) of a sequence X (k) is given by

The DFT pair can also be written as

DFT pair

X (n) X (k)

2. Write the analysis and synthesis equation of DFT (AU DEC 03)

Analysis:

Synthesis:

3. List out the properties of DFT (MU Oct 95,98,Apr 2000)

Linearity and Periodicity

Time reversal of a sequence

Circular time shifting of a sequence

Frequency shifting property

Symmetry property and Convolution property

4. What is Twiddle factor? State the properties of Twiddle factor.

In DFT computation, the term 𝑊𝑁 = 𝑒−𝑗2𝜋/𝑛 is called as phase factor (or) twiddle factor. It is used to

reduce the computational complexity and computational time.

Properties of Twiddle factor:

i) Periodicity property : 𝑊𝑁𝑘+𝑁 = 𝑊𝑁

𝑘

ii) Symmetric property : 𝑊𝑁𝑘+𝑁/2

= −𝑊𝑁𝑘

5. Write two applications of DFT.

The DFT is used for spectral analysis of signals using a digital computer.

The DFT is used to perform filtering operations on signals using digital computer

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6. What is zero padding? What are its uses? (AU DEC 04)

Let us consider a signal with length L given by (𝑛) = 𝑥(0), 𝑥(1) … 𝑥(𝐿 − 1) . The minimum number

of equally spaced frequency points that can be calculated between 0 and 2π without time domain

aliasing is L. The way to improve the frequency resolution is to pad zeros to the signal as 𝑥(𝑛) =𝑥(0), 𝑥(1) … 𝑥(𝐿 − 1), 0,0 … 0. This process is called as zero padding.

Uses of zero padding:

i) We can get better display of the frequency spectrum

ii) With zero padding, the DFT can be used in linear filtering

7. Assume two finite duration sequences x1(n) and x2(n) are linearly combined. What is the DFT of

x3(n)? Let x3(n)=Ax1(n)+Bx2(n)) (MU Oct 95)

Let 𝑿𝟏(𝒌) = ∑ 𝒙𝟏(𝒏)𝒆−𝒋𝟐𝝅𝒌𝒏/𝑵𝑵−𝟏𝒏=𝟎 ; 𝑿𝟐(𝒌) = ∑ 𝒙𝟐(𝒏)𝒆−𝒋𝟐𝝅𝒌𝒏/𝑵𝑵−𝟏

𝒏=𝟎

𝑿𝟑(𝒌) = ∑ 𝒙𝟑(𝒏)𝒆−𝒋𝟐𝝅𝒌𝒏

𝑵

𝑵−𝟏

𝒏=𝟎

= ∑[𝒙𝟏(𝒏) + 𝒙𝟐(𝒏)]𝒆−𝒋𝟐𝝅𝒌𝒏

𝑵 = ∑ 𝒙𝟏(𝒏)𝒆−𝒋𝟐𝝅𝒌𝒏

𝑵 + ∑ 𝒙𝟐(𝒏)𝒆−𝒋𝟐𝝅𝒌𝒏

𝑵

𝑵−𝟏

𝒏=𝟎

𝑵−𝟏

𝒏=𝟎

𝑵−𝟏

𝒏=𝟎

𝑿𝟑(𝒌) = 𝑿𝟏(𝒌) + 𝑿𝟐(𝒌)

8. Define Butterfly diagram. Mention the memory requirement for 2-point butterfly structure.

Butterfly is a portion of the computation that combines the results of smaller discrete Fourier

transforms (DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into sub transforms). The

name "butterfly" comes from the shape of the data-flow diagram in the radix-2 case, as indicated in fig.

9. Distinguish between DFT and DTFT (AU APR 04) (or) What is the relation between DTFT and

DFT

S. No DFT DTFT

1. Obtained by performing sampling operation

in both the time and frequency domains.

Sampling is performed only in time domain.

2. DFT[x (n)] = X (k) DTFT[x (n)] = X (ω)

3. Discrete frequency spectrum Continuous function of ɷ

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10. State the circular time shifting and circular frequency shifting properties of DFT

Circular time reversal : DFT x ((−n))𝑁 = X((−k))𝑁 =X (N-k)

Circular frequency shifting : DFT x ((n − m))𝑁 = e−j2πkm/NX(k)

11. What is FFT? Why FFT is needed? (AU DEC 03) (AU DEC 06) (MU Oct 95,Apr 98)

The FFT is a method for computing the DFT with reduced number of calculations. The computational

efficiency is achieved by divide and conquers approach. This is based on the decomposition of an N –

point DFT into successively smaller DFTs.

12. What are the methods to improve the speed of the computations in DFT? (or) What are the

methods available for finding the FFT? What is the main advantage of FFT?

The DFT computation time can be effectively reduced by incorporating the symmetry and periodicity

properties of twiddle factor into the DFT computation. The fast Fourier transform is an algorithm used

to compute the DFT. It is based on the fundamental principle of decomposing the computation of DFT

of a sequence of length N into successively smaller DFTs.

13. State the computational complexity involved in DFT. (or) How many multiplications and

additions are required to compute N-point DFT using radix-2 FFT? (AU DEC 04)

Direct computation of DFT:

i) No of multiplication required: 𝑁2

ii) No of additions required : N(N-1)

DFT computation using FFT:

i) No of multiplication required: 𝑁𝑙𝑜𝑔2𝑁

ii) No of additions required : 𝑁

2𝑙𝑜𝑔2𝑁

14. What is the speed improvement factor in calculating 64 point DFT of a sequence using direct

computation and FFT algorithm? (or) Calculate the number of multiplications needed in the

calculation of DFT and FFT with 64 point sequence.

Direct computation of DFT:

iii) No of multiplication required = 𝑁2 = 642 = 4096

iv) No of additions required = N(N-1)=64*63= 4032

DFT computation using FFT:

iii) No of multiplication required:=𝑁𝑙𝑜𝑔2𝑁 = 64𝑙𝑜𝑔264 = 384

iv) No of additions required = 𝑁

2𝑙𝑜𝑔2𝑁 =

64

2𝑙𝑜𝑔264 = 192 .

The speed improvement factor in additions= 4032/192=21

The speed improvement factor in multiplications= 4096/384=48.76

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15. What is Radix -2 algorithm?

The FFT algorithm is most efficient in calculating N-point DFT. If the number of output points N can

be expressed as a power of 2, that is N=2𝑀, where M is an integer, then this algorithm is known as

radix-2 FFT algorithm.

16. What is a decimation-in-time algorithm?

Decimation-in-time algorithm is used to calculate the DFT of a N-point sequence. The idea is to break

the N-point sequence into two sequences, the DFTs of which can be combined to give the DFT of

original N-point sequence. Initially the N-point sequence is divided into two N/2 point sequences

𝑥𝑒(𝑛) and 𝑥𝑜(𝑛), which have even and odd members of x (n) respectively. The N/2 point DFTs of

these two sequences are evaluated and combined to give the N-point DFT. Similarly the N/2 point

DFTs can be expressed as a combination of N/4 point DFTs. This process is continued until we are left

with 2 point DFT. This algorithm is called DIT because the sequence x (n) is often split into smaller

subsequences.

17. What is a decimation-in-frequency algorithm?

It is a popular form of the FFT algorithm. In this the output sequence X(k) is divided into smaller and

smaller subsequences , that is why the name decimation in frequency. Initially the input sequence x(n)

is divided into two sequences x1(n) and x2(n) consisting of the first n/2 samples of x(n) and the last N/2

samples of x(n) respectively. This process is continued until we are left with 2 point DFT. This

algorithm is called DIF

18. What are the differences and similarities between DIF and DIT? (or) Compare DIT and DIF

S. No DIT radix – 2 FFT. DIF radix – 2 FFT.

1. The time domain sequence is decimated. The frequency domain sequence is

decimated.

2. When the input is in bit reversed order,

the output will be in normal order and

vice versa.

When the input is in bit normal order, the

output will be in bit reversed order and vice

versa.

3. In each stage of computations, the

phase factors are multiplied before add

and subtract operations.

In each stage of computations, the phase

factors are multiplied after add and subtract

operations.

4. The value of N should be expressed

such that N = 2m and this algorithm

consists of m stages of computations.

The value of N should be expressed such

that N = 2m and this algorithm consists of m

stages of computations.

5. Total number of arithmetic operations is

Nlog2N complex additions and (N/2)

log2N complex multiplications.

Total number of arithmetic operations is

Nlog2N complex additions and (N/2) log2N

complex multiplications.

19. What is overlap save method? Why it is needed?

In this method, we append (M-1) zeros to each data block and perform N point circular convolution

of 𝑥𝑖(𝑛) with h (n). Since, each data block is terminated with M-1 zeros, the last M-1 points from each

output block must be overlapped and added to first M-1 points of the succeeding block. Hence, this

method is called overlap- add method.

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20. What do you mean by “In place‟ computation in DIT-FFT algorithm?

The basic butterfly diagram used in DIT is shown in fig. Here the two lines emerging from two nodes

cross each other and connected to the two nodes on the right hand side. These nodes represent memory

locations. At the input nodes, the inputs are stored. After the outputs are calculated, the same memory

location is used to store the new values in place of the input values. The algorithm uses same location

to store both the input and output sequences are called an ‘in-place’ algorithm.

21. What are the applications of FFT algorithms?

The applications of FFT algorithm include

i) linear filtering

ii) correlation

iii) Spectrum analysis

22. Draw the basic butterfly diagram for radix 2 DIT-FFT and DIF-FFT(AU DEC 07)

Fig (a) Radix-2 DIT-FFT Fig (b) Radix-2- DIF-FFT

23. What is sectioned convolution? (or) What is meant by block convolution? What are the methods

used in sectioned convolution? (or) What is fast convolution? What are the types of Fast

convolution algorithms?

If the data sequence x(n) is of long sequence duration, it is very difficult to obtain the output sequence

y(n) due to limited memory of a digital computer. Therefore, the data sequence is divided into smaller

sections. These sections are processed separately one at a time and combined later to get the output.

This process is said to be sectioned convolution of fast convolution. There are two types: i) overlap-

adds method ii) over lap save method.

24. What is overlap save method? Why it is needed?

In this method, the data sequence is divided into N-point sections 𝑥𝑖(𝑛) . Each section contains the last

M-1 data points of the previous section followed by L new data points to form a data sequence of

length N=L+M-1. If we take N-point circular convolution of 𝑥𝑖(𝑛) with h(n), the first M-1 points will

results from aliasing the remaining points are the required convolution. Hence, we discard the first (M-

1) points of filter section. This process is repeated for all sections and the filtered sections are combined

for final result.

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25. Distinguish between linear convolution and Circular Convolution. (MU Oct 96,Oct 97,Oct 98)

S. No Linear Convolution Circular Convolution

1. If x(n) is a sequence of L number of

samples and h(n) with m number of

samples, after convolution y(n) will

contain N = L + M – 1 samples.

If x(n) is a sequence of L number of

samples and h(n) with m number of

samples, after convolution y(n) will

contain N = Max(L,M) samples

2. Linear convolution can be used to find

the response of a linear filter.

Circular convolution can be used to find

the response of a linear filter

3. Zero padding is not necessary to find the

response of a linear filter.

Zero padding is necessary to find the

response of a linear filter.

26. Compare overlap- add and overlap-save methods of sectioned convolution.

S.

No

Overlap Add method Overlap Save method

1 Linear convolution of each section of

longer sequence with smaller sequence is

performed.

Circular convolution of each section of longer

sequence with smaller sequence is performed.

(After converting them to the size of output

sequence)

2 Zero Padding is not required.

Zero Padding is required to convert the input

sequences to the size of the output sequence

3 Overlapping of samples of input sections is

not required.

The N2-1 samples of an input section of longer

sequence are overlapped with next input

section.

4 The overlapped samples in the output of

sectioned convolutions are added to get the

overall output.

Depending on the method of overlapping the

input samples, either the last N2-1 samples or

the first N2-1 samples of the output sequence

of each sectioned convolutions are discarded.

27. The first five coefficients of X(K)=1, 0.2+5j, 2+3j, 2 ,5 Find the remaining coefficients

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UNIT III IIR FILTER DESIGN

1. Define an IIR filter.

The filters designed by considering all the infinite samples of impulse response are called IIR

filters. The impulse response is obtained by taking inverse Fourier transform of ideal frequency

response.

2. Compare IIR and FIR filters.

S.

No

IIR Filter FIR Filter

1 All the infinite samples of impulse response

are considered

Only N samples of impulse response are

considered

2 The impulse response cannot be directly

converted to digital transfer function

The impulse response can be directly

converted to digital transfer function

3 The design involves design of analog filter and

then transforming analog filter to digital filter

The digital filter can be directly designed

to achieve the desired specifications

4 The specifications include the desired

characteristics for magnitude response only

The specifications include the desired

characteristics for both magnitude and

phase response

5 Linear phase characteristics cannot be

achieved

Linear phase filters can be easily

designed

3. What are the requirements for an analog filter to be stable and causal?

o The analog filter transfer function Ha(s) should be a rational function of s and the coefficients of

s should be real.

o The poles should lie on the left half of s – plane.

o The number of zeroes should be less than or equal to number of poles.

4. What are the requirements for a digital filter to be stable and causal?

o The analog filter transfer function H (z) should be a rational function of z and the coefficients of

z should be real.

o The poles should lie inside the unit circle in z - plane.

o The number of zeroes should be less than or equal to number of poles.

5. Compare the digital and analog filter.

S. No Digital Filter Analog Filter

1 Operates on digital samples of the signal Operates on analog signals

2 It is governed by linear difference equation It is governed by linear differential

equation

3

It consists of adders, multipliers and delays

implemented in digital logic.

It consists of electrical components like

resistors, capacitors and inductors

4. Filter coefficients are designed to satisfy the

desired frequency response.

Approximation problem is solved to

satisfy the desired frequency response

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6. What are the advantages and disadvantages of digital filters?

Advantages of digital filters

o High thermal stability due to absence of resistors, capacitors and inductors.

o The performance characteristics like accuracy, dynamic range, stability and tolerance can be

enhanced by increasing the length of the registers.

o The digital filters are programmable.

o Multiplexing and adaptive filtering are possible.

7. Disadvantages of digital filters

o The bandwidth of the discrete signal is limited by the sampling frequency.

o The performance of the digital filter depends on the hardware used to implement the filter.

8. What is impulse invariant transformation?

The transformation of analog filter to digital filter without modify the impulse response of the filter

is called impulse invariant transformation. In this transformation the impulse response of the digital

filter will be sampled version of the impulse response of the analog filter.

9. What is bilinear transformation?

The bilinear transformation is conformal mapping that transforms the s – plane to z – plane. In this

mapping the imaginary axis of s – plane is mapped into the unit circle in the z – plane, the left half

of s – plane is mapped into interior of unit circle in z – plane and the right half of s – plane is

mapped into exterior of unit circle in z – plane. The bilinear mapping is a one to one mapping and it

is accomplished when

10. What is frequency warping?

In bilinear transformation, the relation between analog and digital frequencies is nonlinear. This

non-linear relationship introduces distortion in frequency axis, when the s – plane is mapped into z

– plane using bilinear transformation. It is called frequency warping.

11. What are the advantages and disadvantages of bilinear transformation?

Advantages of bilinear transformation

o The bilinear transformation is one –to-one mapping.

o There is no aliasing and so the analog filter need not have a band limited frequency

response.

o The effect of warping on amplitude response can be eliminated by pre-warping the

analog filter.

o It can be used to design digital filters with prescribed magnitude response with piecewise

constant values.

Disadvantages of bilinear transformation

o The nonlinear relationship between analog and digital frequencies introduces frequency

distortion which is called frequency warping.

o Using bilinear transformation, a linear phase analog filter cannot be transformed to linear

phase digital filter.

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12. What is pre-warping? Why it is employed.

In IIR filter design using bilinear transformation, the conversion of the specified digital frequencies

to analog frequencies is called pre-warping. It is necessary to eliminate the effect of warping on

amplitude response.

13. Compare the impulse invariant and bilinear transformations

S.

No Impulse Invariant Transformation Bilinear transformation

1 It is many-to-one mapping It is one-to-one mapping

2 The relation between analog and

digital frequency is linear

The relation between analog and digital

frequency is non-linear

3 To prevent the problem of aliasing the

analog filters should be band limited

Is no problem of aliasing and so the

analog filters need not be band limited

4

The magnitude and phase response of

analog filter can be preserved by

choosing low sampling time or high

sampling frequency

Due to the effect of warping, the phase

response of analog filters cannot be

preserved. But the magnitude response

can be preserved by pre-warping

14. What is Butterworth approximation?

In Butterworth approximation, the error function is selected such that the magnitude is maximally

flat in the origin (i.e., at Ω = 0) and monotonically decreasing with increasing Ω.

15. Write the properties of Butterworth filter.

o The Butterworth filters are all pole designs.

o At the cut-off frequency Ωc the magnitude of normalized Butterworth filter is 1/√2.

o The filter order ‘n’ completely specifies the filter and as the value of N increases the

magnitude response approaches the ideal response.

16. What is Chebyshev approximation?

In Chebyshev approximation, the approximation function is selected such that the error is

minimized over a prescribed band of frequencies.

17. What is type – I Chebyshev approximation?

In type – I Chebyshev approximation, the error function is selected such that, the magnitude

response is equi-ripple in the pass band and monotonic in the stop band.

18. What is type – II Chebyshev approximation?

In type – II Chebyshev approximation, the error function is selected such that, the magnitude

response is monotonic is pass band and equi-ripple in stop band. The type – II magnitude response

is called inverse Chebyshev response.

19. Write the properties of Chebyshev type – I filter.

o The magnitude response is equi-ripple in the pass band and monotonic in the stop band.

o The Chebyshev type – I filters are all pole designs.

o The normalized magnitude function has a value of 1/√1+€ at the cut-off frequency Ωc

o The magnitude response approaches the ideal response as the value of N increases.

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20. What are the different types of structures for realization of IIR systems?

The different types of structures for realization of IIR system are.

o Direct – form I structure,

o Direct – form II structure,

o Transposed Direct – form II structure,

o Cascade form structure,

o Parallel form structure,

o Lattice – ladder structure.

21. Why an impulse invariant transformation is not considered to be one-to-one?

In impulse invariant transformation any strip of width 2π/T in the s-plane for values of s-plane in

the range (2k-1)/T ≤ Ω≤ (2k-1) π/T is mapped into the entire z-plane. The left half of each strip in

s-plane is mapped into the interior of unit circle in z-plane, right half of each strip in s-plane is

mapped into the exterior of unit circle in z-plane and the imaginary axis of each strip in s-plane is

mapped on the unit circle in z-plane. Hence the impulse invariant transformation is many-to-one.

22. Compare Butterworth and Chebyshev filters

S.

No

Butterworth filters Chebyshev filters

1 The magnitude response |H(jw)| of the

butter-worth filter decreases with increase

in frequency from 0 to infinity

The magnitude response of the Chebyshev filter

fluctuates or show ripples in the pass band and

stop band depending on the type of the filter.

2 The width of the transition band is more The width of the transition band is less

3 The poles of a Butterworth filter lies only

on a circle

The poles of a Butterworth filter lies on an

ellipse

23. Convert the following analog transfer function into digital using impulse invariant

transformation.

Solution:

.

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24. What are the properties of Butterworth filter?

Properties of Butterworth filter

25. Give the expressions for various analog frequency transformation ( or) What is analog

frequency transformation?

If we wish to design a high pass or band pass or band stop filter, it is a simple method to take a low

pass prototype filter (butter worth, chebyshev) and perform a frequency transformation. This

process is called as analog frequency transformation. Various frequency transformations in analog

domain are listed below

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UNIT IV FIR FILTER DESIGN

1. What is FIR filter?

The specifications of the desired filter will be given in terms of ideal frequency response Hd(w).

The impulse response hd(n) of the desired filter can be obtained by inverse Fourier transform of

Hd(w),which consists of infinite samples. The filters designed by selecting finite number of samples

of impulse response are called FIR filters.

2. What are the different types of filters based on impulse response?

Based on impulse response the filters are of two types 1. IIR filter 2. FIR filters

The IIR filters are of recursive type, whereby the present output sample depends on the present

input, past input samples and output samples.

The FIR filters are of non recursive type, whereby the present output sample depends on the

present input, and previous output samples.

3. What are the different types of filter based on frequency response?

The filters can be classified based on frequency response. They are i) Low pass filter ii) High pass

filter iii) Band pass filter iv) Band reject filter.

4. Distinguish between FIR and IIR filters.

S. No FIR filter IIR filter

1 These filters can be easily designed to

have perfectly linear phase

These filters do not have linear phase.

2 FIR filters can be realized recursively

and non-recursively

IIR filters can be realized recursively

3 Greater flexibility to control the shape

of their magnitude response

Less flexibility, usually limited to kind

of filters

4 Errors due to round-off noise are less

severe in FIR filters, mainly because

feedback is not used

The round-off noise in IIR filters are

more

5. What are the techniques of designing FIR filters? (or) List the well-known design techniques

of linear phase FIR filters.

There are three well-known design techniques of linear phase FIR filters. They are

i. Fourier series method and window method

ii. Frequency sampling method.

iii. Optimal filter design methods.

6. What is the reason that FIR filter is always stable?

FIR filter is always stable because all its poles are at origin.

7. What are the properties of FIR filter?

FIR filter is always stable.

A realizable filter can always be obtained.

FIR filter has a linear phase response.

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8. How phase distortion and delay distortions are introduced?

The phase distortion is introduced when the phase characteristics of a filter is not linear within the

desired frequency band. The delay distortion is introduced when the delay is not constant within the

desired frequency range.

9. Write the steps involved in FIR filter design.

Choose the desired (ideal) frequency response Hd(w). Take inverse Fourier transform of Hd(w) to get hd(n). Convert the infinite duration hd(n) to finite duration h(n). Take Z-transform of h(n) to get the transfer function H(z) of the FIR filter.

10. What are the advantages and disadvantages of FIR filters?

Advantages

Linear phase FIR filter can be easily designed.

Efficient realization of FIR filter exist as both recursive and non-recursive structures.

FIR filters are always stable.

The round-off noise can be made small in non-recursive realization of FIR filters.

Disadvantages

The duration of impulse response should be large to realize sharp cut-off filters.

The non-integral delay can lead to problems in some signal processing applications.

11. What is the necessary and sufficient condition for the linear phase characteristic of an FIR

filter?

The necessary and sufficient condition for the linear phase characteristic of an FIR filter is that the

phase function should be a linear function of w, which in turn requires constant phase and group

delay.

12. What are the conditions to be satisfied for constant phase delay in linear phase FIR filters? (or) How

constant group delay & phase delay is achieved in linear phase FIR filters?

The conditions for constant phase delay are

Phase delay, α = (N-1)/2 (i.e., phase delay is constant)

Group delay, β = π/2 (i.e., group delay is constant)

Impulse response, h (n) = -h (N-1-n) (i.e., impulse response is anti-symmetric)

13. What are the possible types of impulse response for linear phase FIR filters?

There are four types of impulse response for linear phase FIR filters Symmetric impulse response when N is odd.

Symmetric impulse response when N is even.

Anti-symmetric impulse response when N is odd.

Anti-symmetric impulse response when N is even.

14. What is Gibb’s phenomenon (or Gibb’s Oscillation)?

In FIR filter design by Fourier series method the infinite duration impulse response is truncated to

finite duration impulse response. The abrupt truncation of impulse response introduces oscillations

in the pass band and stop band. This effect is known as Gibb’s phenomenon (or Gibb’s Oscillation).

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15. What are the desirable characteristics of the frequency response of window function?

The desirable characteristics of the frequency response of window function are

o The width of the main-lobe should be small and it should contain as much of the total energy

as possible.

o The side-lobes should decrease in energy rapidly as w tends to π.

16. Write the procedure for designing FIR filter using frequency-sampling method.

o Choose the desired (ideal) frequency response Hd(w).

o Take N-samples of Hd(w) to generate the sequence

o Take inverse DFT of to get the impulse response h(n).

o The transfer function H(z) of the filter is obtained by taking z-transform of impulse response.

17. What are the drawback in FIR filter design using windows and frequency sampling method?

How it is overcome?

The FIR filter design using windows and frequency sampling method does not have Precise control

over the critical frequencies such as wp and ws. This drawback can be overcome by designing FIR

filter using Chebyshev approximation technique. In this technique an error function is used to

approximate the ideal frequency response, in order to satisfy the desired specifications.

18. Write the characteristic features of rectangular window.

o The main lobe width is equal to 4π/N.

o The maximum side lobe magnitude is –13dB.

o The side lobe magnitude does not decrease significantly with increasing w.

19. List the features of FIR filter designed using rectangular window.

The width of the transition region is related to the width of the main lobe of window

spectrum.

Gibb’s oscillations are noticed in the pass band and stop band.

The attenuation in the stop band is constant and cannot be varied.

20. Why Gibb’s oscillations are developed in rectangular window and how it can be eliminated or

reduced?

The Gibb’s oscillations in rectangular window are due to the sharp transitions from 1 to 0 at the

edges of window sequence. These oscillations can be eliminated or reduced by replacing the sharp

transition by gradual transition. This is the motivation for development of triangular and cosine

windows.

21. List the characteristics of FIR filters designed using windows.

o The width of the transition band depends on the type of window.

o The width of the transition band can be made narrow by increasing the value of N where N is

the length of the window sequence.

o The attenuation in the stop band is fixed for a given window, except in case of Kaiser window

where it is variable.

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22. Compare the rectangular window and Hanning window.

S.No Rectangular window Hanning Window

1 The width of main lobe in window

spectrum is 4π/N

The width of main lobe in window

spectrum is 8π/N

2 The maximum side lobe magnitude

in window spectrum is –13dB.

The maximum side lobe magnitude in

window spectrum is –31dB.

3 In window spectrum the side lobe

magnitude slightly decreases with

increasing w.

In window spectrum the side lobe

magnitude decreases with increasing w.

4 In FIR filter designed using

rectangular window the minimum

stop band attenuation is 22dB

In FIR filter designed using hanning

window the minimum stop band attenuation is 44dB

23. Compare the rectangular window and hamming window.

S.No Rectangular window Hamming Window

1 The width of main lobe in window

spectrum is 4π/N

The width of main lobe in window

spectrum is 8π/N

2 The maximum side lobe magnitude

in window spectrum is –13dB.

The maximum side lobe magnitude in

window spectrum is –41dB

3 In window spectrum the side lobe

magnitude slightly decreases with

increasing w.

In window spectrum the side lobe

magnitude remains constant

4 In FIR filter designed using

rectangular window the minimum

stop band attenuation is 22dB

In FIR filter designed using hamming

window the minimum stop band

attenuation is 44dB

24. Write the characteristic features of hanning window spectrum.

The main lobe width is equal to 8π/N.

The maximum side lobe magnitude is –41dB

The side lobe magnitude remains constant for increasing w.

25. What is the mathematical problem involved in the design of window function?

The mathematical problem involved in the design of window function (or sequence) is that of

finding a time-limited function whose Fourier Transform best approximates a band limited

function. The approximation should be such that the maximum energy is confined to main lobe for

a given peak side lobe amplitude.

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26. List the desirable features of Kaiser Window spectrum.

o The width of the main lobe and the peak side lobe are variable.

o The parameter α in the Kaiser Window function is an independent variable that can be varied to

control the side lobe levels with respect to main lobe peak.

o The width of the main lobe in the window spectrum can be varied by varying the length N of the

window sequence.

27. Compare the hamming window and Kaiser window.

S.

No

Kaiser Window Hamming Window

1 The width of main lobe in window

spectrum depends on the values of α

& N

The width of main lobe in window

spectrum is 8π/N

2 The maximum side lobe magnitude

with respect to peak of main lobe is

variable using the parameter α

The maximum side lobe magnitude in

window spectrum is –41dB

3 In window spectrum the side lobe

magnitude decreases with increasing

w.

In window spectrum the side lobe

magnitude remains constant

4 In FIR filter designed using Kaiser

window the minimum stop band

attenuation is variable and depends

on the value of α.

In FIR filter designed using hamming

window the minimum stop band

attenuation is 44dB

28. What are the quantization methods? (or) Define the terms i)rounding ii) truncation

Quantization methods: Truncation -- Truncation is a process of discarding all bits less significant than LSB that is retained

Rounding -- Rounding a number to b bits is accomplished by choosing a rounded result as the b bit

number closest number being unrounded.

29. What is input quantization error?

The filter coefficients are computed to infinite precision in theory. But in digital computation the

filter coefficients are represented in binary and are stored in registers. If a b bit register is used the

filter coefficients must be rounded or truncated to b bits, which produces an error.

30. What is product quantization error?

The product quantization errors arise at the output of the multiplier. Multiplication of a b bit data

with a b bit coefficient results a product having 2b bits. Since a b bit register is used the multiplier

output will be rounded or truncated to b bits which produce the error.

31. What is zero limit cycle oscillation? (or) What is meant by limit cycle oscillation?

For an IIR filter, implemented with infinite precision arithmetic the output should approach zero in

the steady state if the input is zero .however the non linearity due to the finite precision arithmetic

operation often cause periodic oscillations to occur in the output. Such oscillations in recursive

systems are called zero input limit cycle oscillations.

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32. Define dead band. Give its expression (or) Define "dead band" of the filter

The limit cycle occur as a result of quantization effect in multiplication. The amplitudes of the

output during a limit cycle are confined to a range of values called the dead band of the filter.

33. What is finite word length effect? (or) what are the sources of finite word length ? (or)What

are the three-quantization errors to finite word length registers in digital filters?

1. Input quantization error

2. Coefficient quantization error

3. Product quantization error

34. What is overflow error? What are the methods to eliminate the overflow? (or)What is

overflow oscillation? What are the methods used to prevent overflow?

In case of adding two fixed-point arithmetic numbers the sum exceeds the word size available to

store the sum which cause overflow error. This overflow caused make the filter output to oscillate

between maximum amplitude limits. Such limit cycles have been referred to as over flow

oscillations

There are two methods used to prevent overflow

1. Saturation arithmetic 2. Scaling

35. Why rounding is preferred to truncation in realizing digital filter?

1. The quantization error due to rounding is independent of type of arithmetic.

2. The mean of rounding error is zero.

3. The variance of the rounding error signal is low.

36. Compare the fixed point and floating point arithmetic.

S.No Fixed point arithmetic.

Floating point arithmetic.

1 Fast operation

Slow operation

2 Relatively economical More expensive because of costlier

hardware.

3 Small dynamic range Increased dynamic range

4 Round off error occur

only for addition

Round off error occur with both

addition and multiplication.

5 Overflow occur in

addition Overflow occur in addition

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UNIT V APPLICATIONS

1. Define multi rate digital signal processing. (or) What is the need for multirate signal

processing?

In multirate digital signal processing the sampling rate of a signal is changed in order to increase

the efficiency of various signal processing operations. Decimation, or down-sampling, reduces the

sampling rate, whereas expansion, or up-sampling, followed by interpolation increases the sampling

rate. In multirate DSP systems, sample rates are changed (or are different) within the system.

Advantages: o Reduced computational complexity

o Reduced transmission data rate.

2. What is adaptive equalization? (or) What is adaptive equalizer?

Adaptive equalization is the technique used to reliably transmit data through a communication

channel. An adaptive equalizer is an equalizer that automatically adapts to time-varying properties

of the communication channel. It is frequently used with coherent modulations such as phase shift

keying, mitigating the effects of multipath propagation and Doppler spreading.

3. What do you mean by speech compression? List various voice compression and coding

techniques.

The process of encoding digital speech to take up less storage space and transmission bandwidth is

called as speech compression.

PCM (Pulse code modulation)

ADPCM (Adaptive differential pulse code modulation)

Linear predictive coding (LPC)

code excited linear prediction compression (CELP)

Multi-pulse, multilevel quantization (MP-MLQ)

4. Define up sampling. Give its significance.

Up sampling is the process of increasing the sample rate of a digital audio signal. For example, the

sample rate of a CD is 44.1 kHz, which means the analog signal is sampled 44,100 times per

second. Up sampling by a factor of two increases the sample rate from 44.1 kHz to 88.2 kHz,

effectively doubling the number of samples available. It widens frequency response.

5. What is adaptive filter? List out the applications of adaptive filtering. ( or) State few

applications of adaptive filter.

An adaptive filter is a computational device that iteratively models the relationship between the

input and output signals of the filter. An adaptive filter self-adjusts the filter coefficients according

to an adaptive algorithm.

Applications:

Noise cancellation

Signal prediction

Adaptive feedback cancellation

Echo cancellation

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6. What is linear predictive coding? (or) What is basic idea of linear predictive coding?

Linear predictive coding (LPC) is a tool used mostly in audio signal processing and speech

processing for representing the spectral envelope of a digital signal of speech in compressed form,

using the information of a linear predictive model. In these cases, a linear predictor can be used to

model the signal correlations for a short block of data in such a way as to reduce the number of bits

needed to represent the signal waveform.

7. Define adaptive LMS algorithm.

The least mean squares (LMS) algorithms adjust the filter coefficients to minimize the cost

function. Compared to recursive least squares (RLS) algorithms, the LMS algorithms do not

involve any matrix operations. Therefore, the LMS algorithms require fewer computational

resources and memory than the RLS algorithms. The implementation of the LMS algorithms also is

less complicated than the RLS algorithms.

8. A signal x (n) = 6, 1, 5, 7, 2, 1. Find x(n/2), x(2n) and x(3n).

9. Compare traditional digital filter with an adaptive filter. (or) How adaptive filter is differed

from conventional filter?

S. No Traditional digital filter Adaptive filter

1 A traditional digital filter has only one

input signal x(n) and one output signal

y(n).

An adaptive filter requires an additional

input signal d(n) and returns an additional

output signal e(n).

2 The filter coefficients of a traditional

digital filter do not change over time

The coefficients of an adaptive filter change

over time

3 Adaptive filters have a self-learning

ability.

Traditional digital filters do not have self-

learning ability.

10. Prove that up/down sampling by a factor M is time varying system.

Down-sampling by M is a linear, periodically time varying operator with period M.

Proof:

i) Linearity:

𝐷𝑀(𝛼 𝑋1[𝑛] = 𝛽 𝑋2[𝑛]) = 𝛼 𝑋1[𝑀𝑛] + 𝛽 𝑋2[𝑀𝑛] = 𝛼 𝐷𝑀[𝑋1[𝑛]] + 𝛽 𝐷𝑀[𝑋2[𝑛]] It clearly shows that 𝐷𝑀 is a linear operator.

ii) Time-varying property:

It has a periodically time-varying property because if a sequence is shifted by M, its down-sampled

version is shifted by 1.

If (𝑛) = 𝐷𝑀(𝑥[𝑛]) , then 𝑦(𝑛) = 𝐷𝑀(𝑥[𝑛 − 𝑘𝑀]) = 𝑦[𝑛 − 𝑘].

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11. List various special audio effects that can be implemented digitally.

Flanging: It is a time-based audio effect created by mixing a signal with a delayed version of itself.

Reverberation: Reverberation is a property of sound in an enclosed space such as an auditorium.

Synthetic Stereo: One way of creating the sensation of stereo from a mono source is to add a

delayed version of the signal to itself.

12. Define sub band coding. What are the sub band coding techniques?

In signal processing, Sub-band coding (SBC) is any form of transform coding that breaks a signal

into a number of different frequency bands and encodes each one independently. This

decomposition is often the first step in data compression for audio and video signals.

13. What is QMF? (or) Define Quadrature mirror filters.

In digital signal processing, a Quadrature mirror filter is a filter whose magnitude response is

mirror image about π/2 of that of another filter. Together these filters are known as the Quadrature

Mirror Filter pair. A filter 𝐻1(𝑧)will be Quadrature mirror filter of 𝐻0(𝑧), if 𝐻1(𝑧) = 𝐻0(−𝑧). The

filter responses are symmetric about Ω=π/2

14. What are the two techniques of sampling rate conversion?

o D/A conversion and re-sampling at required rate.

o Sampling rate conversion in digital domain (multi-rate processing)

15. What are the applications of sampling rate conversion? o Narrow band filters.

o Quadrature mirror filters.

o Digital filter banks.

16. What is meant by decimation? Write the decimation equation?

Decimation by a factor D means to reduce the sampling rate by a factor D. It is also called down

sampling. When v (n) is down sampled by the factor D, y (m) = v(mD)

17. What is meant by interpolation? Write the interpolation equation?

Interpolation by a factor I, means to increase the sampling rate by a factor I. It is also called as up

sampling by I.

18. Define speech compression and decompression.

Speech analysis by a Vocoder becomes the compression and synthesis by a Vocoder becomes

decompression. Vocoder extracts the spectral envelope of speech and information regarding voicing

and pitch. This data is coded and transmitted. The synthesizer generates speech from the received

data.

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19. Write the principle of adaptive filters.

The coefficients of the filter are changed automatically according to the changes in input signal.

This means the filtering characteristics of the adaptive filter are changed or adapted according to the

changes in input signal.

20. How the image enhancement is achieved using DSP?

o Local neighbourhood operations as in convolution.

o Transform operations as in DFT.

o Mapping operations as in pseudo colouring and gray level mapping.

21. List the different methods of image enhancement. (or) What are the various enhancement

techniques in image processing? (or) Give any two image enhancement methods.

o Contrast and edge enhancement.

o Pseudo colouring.

o Noise filtering.

o Sharpening.

o Magnifying.

22. What are the applications of image enhancement?

o Feature extraction in an image.

o Image analysis.

o Visual information display.