Dsp_ece_5th Sem (2mark Q&A)

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Prepared by Prof. U. Vinothkumar AP / ECE/ Dr.NGPIT

Dr. N.G.P.INSTITUTE OF TECHNOLOGY, COIMBATORE - 641048.

Department of Electronics & Communication Engineering

Question Bank

Anna University, Chennai.

EC6502 - PRINCIPLES OF DIGITAL SIGNAL PROCESSING

Prepared by,

Prof. U. Vinothkumar, AP/ECE/Dr.N.G.P.IT

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UNIT - 1 DISCRETE FOURIER TRANSFORM

Syllabus: Discrete Signals and Systems- A Review – Introduction to DFT – Properties

of DFT – Circular Convolution – Filtering methods based on DFT – FFT Algorithms –Decimation in time Algorithms, Decimation in frequency Algorithms –Use of FFT in Linear Filtering.

Two mark questions: 1. Define Signal.

Signal is a physical quantity that varies with respect to time, space or any other independent variable.

(Or) It is a mathematical representation of the system Eg y(t) = t. and x(t)= sin t. 2. Define system. (NOV-2004)

A set of components that are connected together to perform the particular task. E.g. Filters

(Or) A System is defined as a physical device that generates a response or an output

signal, for a given input signal.

3. State the classification of discrete time signals. The types of discrete time signals are

* Energy and power signals * Periodic and A periodic signals * Symmetric (Even) and Ant symmetric (Odd) signals

4. State the classification of discrete time system. (APR-2006) They types of discrete time systems are * Static and Dynamic systems * Causal and non-causal systems * Linear and non-linear systems * Time variant and time in-variant systems

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5. Define Discrete-time system.

A discrete time system is one which operates on a discrete-time signal and produces a discrete-time output signal. If the input and output of discrete-time system are x(n) and y(n), then we can write y(n)= T[x(n)]. 6. Define Discrete-time signal. (APR-2008)

The signal that are defined at discrete instants of time are known as discrete-time signals. The discrete-time signals are continuous in amplitude and discrete in time. They are denoted by x(n). 7. Give some applications of DSP?

* Speech processing – Speech compression & decompression for voice storage system * Communication – Elimination of noise by filtering and echo cancellation. * Bio-Medical – Spectrum analysis of ECG, EEG etc.

8. Define sampling theorem.

A continuous time signal can be represented in its samples and recovered back if the sampling frequency Fs 2B. Here ‘Fs’ is the sampling frequency and ‘B’ is the maximum frequency present in the signal. 9. What are the properties of convolution?

* Commutative property x(n) * h(n) = h(n) * x(n) * Associative property [x(n) * h1(n)]*h2(n) = x(n)*[h1(n) * h2(n)] * Distributive property x(n) *[ h1(n)+h2(n)] = [x(n)*h1(n)]+[x(n) * h2(n)]

10. Define DFT. (APR-2006)

It is a finite duration discrete frequency sequence, which is obtained by sampling one period of Fourier transform. Sampling is done at N equally spaced points over the period extending from w=0 to 2л.

DFT is defined as X(w)= x(n)e-jwn. Here x(n) is the discrete time sequence X(w) is the fourier transform of x(n). 11. Define Twiddle factor.

The Twiddle factor is defined as WN=e-j2 /N

12. Define Zero padding. The method of appending zero in the given sequence is called as Zero padding.

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13. State circular convolution. This property states that multiplication of two DFT is equal to circular

convolution of their sequence in time domain.

14. State parseval’s theorem. (DEC-2006) Consider the complex valued sequences x(n) and y(n).If x(n)y*(n)=1/N

X(k)Y*(k) 15. List the properties of DFT.

Linearity, Periodicity, Circular symmetry, symmetry, Time shift, Frequency shift, complex conjugate, convolution, correlation and Parseval’s theorem. 16. What is the disadvantage of direct computation of DFT?

For the computation of N-point DFT, N2 complex multiplications and N[N-1] Complex additions are required. If the value of N is large than the number of computations will go into lakhs. This proves inefficiency of direct DFT computation. 17. What is the way to reduce number of arithmetic operations during DFT

computation? Number of arithmetic operations involved in the computation of DFT is

greatly reduced by using different FFT algorithms as follows. 1. Radix-2 FFT algorithms. -Radix-2 Decimation in Time (DIT) algorithm. -

Radix-2 Decimation in Frequency (DIF) algorithm. 2. Radix-4 FFT algorithm.

18. What is the computational complexity using FFT algorithm?

1. Complex multiplications = N/2 log2N 2. Complex additions = N log2N

19. Why FFT is needed? (MU Oct’95, Apr’98)

The direct evaluation of the DFT using the formula requires N2 complex multiplications and N (N-1) complex additions. Thus for reasonably large values of N (inorder of 1000) direct evaluation of the DFT requires an inordinate amount of computation. By using FFT algorithms the number of computations can be reduced. For example, for an N-point DFT, The number of complex multiplications required using FFT is N/2log2N. If N=16, the number of complex multiplications required for direct evaluation of DFT is 256, whereas using DFT only 32 multiplications are required.

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20. What is a decimation-in-time algorithm? (DEC-2006)

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 the original N-point sequence. Initially the N-point sequence is divided into two N/2-point sequences xe(n) and x0(n), which have the 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 till we left with 2-point DFT. This algorithm is called Decimation-in-time because the sequence x(n) is often splitted into smaller sub sequences. 21. What are the differences and similarities between DIF and DIT algorithms?

Differences: 1. For DIT, the input is bit reversal while the output is in natural order, whereas for DIF, the input is in natural order while the output is bit reversed. 2. The DIF butterfly is slightly different from the DIT butterfly, the difference being that the complex multiplication takes place after the add-subtract operation in DIF.

Similarities: Both algorithms require same number of operations to compute the DFT. Bot algorithms can be done in place and both need to perform bit reversal at some place during the computation. 22. What are the applications of FFT algorithms?

1. Linear filtering 2. Correlation 3. Spectrum analysis

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

In this the output sequence X (K) is divided into two N/2 point sequences and each N/2 point sequences are in turn divided into two N/4 point sequences. 24. Distinguish between DFT and DTFT. (NOV-2008)

S.No. DFT DTFT 1. 2.

Obtained by performing sampling operation in both the time and frequency domains. Discrete frequency spectrum

Sampling is performed only in time domain. Continuous function of ω

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25. What is the speed improvement factor in calculating 64-point DFT of a sequence using direct computation and FFT algorithms?

The number of complex multiplications required using direct computation is N2 = 642 = 4096.

The number of complex multiplications required using FFT is [N/2] log2N = [64/2] log264 = 192.

Speed improvement factor = [4096/192] = 21.33 26. Calculate the number of multiplications needed in the calculation of DFT

using FFT algorithm with 32-point sequence. (DEC-2009) For N-point DFT the number of complex multiplications needed using FFT

algorithm is [N/2] log2N. For N =32, the number of complex multiplications is equal to [32/2]

log232=16 x5 = 80. 27. How many multiplications and additions are required to compute N-point

DFT using radix-2 FFT? The number of multiplications and additions required to compute N-point

DFT using radix-2 FFT are Nlog2N and [N/2] log2N respectively. 28. What is meant by radix-2 FFT?

The FFT algorithms 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. 29. What is bin spacing? (NOV-2008)

The N-point DFT of x(n) is given by,

X(k) = 푥(푛)푒 / = 푥 푊푁

Where, WNnk = (e-j2π)[nk/N] is the phase factor or twiddle factor.

The phase factors are equally spaced around the unit circle at frequency increments of Fs/N where Fs is the sapling frequency of the time domain signal. This frequency increment or resolution is called bin spacing. (The x (k) consists of N-numbers of frequency samples whose discrete frequency locations are given by fk= kFs/N, for k=0, 1, 2, …..N-1).

30. Arrange the 8-point sequence, x(n) = 1,2,3,4,-1,-2,-3,-4 in bit reversed order.

The x(n) in bit reversed order= 1,-1,3,-3,2,-2,4,-4.

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UNIT - 2

IIR Filter Design Syllabus:

Structures of IIR – Analog filter design – Discrete time IIR filter from analog filter – IIR filter design by Impulse Invariance, Bilinear transformation, Approximation of derivatives – (LPF, HPF, BPF, BRF) filter design using frequency translation.

Two mark questions: 1. Define IIR filter? (NOV-2010)

IIR filter has Infinite Impulse Response.

2. What are the various methods to design IIR filters? * Approximation of derivatives * Impulse invariance * Bilinear transformation.

3. Which of the methods do you prefer for designing IIR filters? Why? Bilinear transformation is best method to design IIR filter, since there is no

aliasing in it.

4. What is the main problem of bilinear transformation? Frequency warping or nonlinear relationship is the main problem of bilinear

transformation.

5. What is pre-warping? (APR-2010) Pre-warping is the method of introducing nonlinearly in frequency

relationship to compensate warping effect. 6. 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.

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7. What is Bi-linear 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 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. 8. How the order of the filter affects the frequency response of Butterworth

filter. The magnitude response of butterworth filter is shown in figure, from which

it can be observed that the magnitude response approaches the ideal response as the order of the filter is increased. 9. What is the importance of poles in filter design? (APR-2008)

The stability of a filter is related to the location of the poles. For a stable analog filter the poles should lie on the left half of s-plane. For a stable digital filter the poles should lie inside the unit circle in the z-plane. 10. How analog poles are mapped to digital poles in impulse invariant

transformation? In impulse invariant transformation the mapping of analog to digital poles

are as follows, * The analog poles on the left half of s-plane are mapped into the interior of

unit circle in z-plane. * The analog poles on the imaginary axis of s-plane are mapped into the unit

circle in the z-plane. * The analog poles on the right half of s-plane are mapped into the exterior

of unit circle in z-plane. 11. What is impulse invariant transformation? (APR-2009)

The transformation of analog filter to digital filter without modifying the impulse response of the filter is called impulse invariant transformation. 12. Where the jΩ axis of s-plane is mapped in z-plane in bilinear

transformation? The j Ω axis of s-plane is mapped on the unit circle in z-plane in bilinear

transformation

13. State the frequency relationship in bilinear transformation? Ω = (2/T) tan (ω/2)

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14. Compare the digital and analog filter. (ARP-2008)

Digital filter

Analog filter

i) Operates on digital samples of

the signal. ii) It is governed by linear

difference equation. iii) It consists of adders,

multipliers and delays implemented in digital logic.

iv) In digital filters the filter coefficients are designed to satisfy the desired frequency response.

i) Operates on analog signals. ii) It is governed by linear

difference equation. iii) It consists of electrical

components like resistors, capacitors and inductors.

iv) In digital filters the approximation problem is solved to satisfy the desired frequency response.

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

Advantages of digital filters High thermal stability due to absence of resistors, inductors and capacitors. Increasing the length of the registers can enhance the performance

characteristics like accuracy, dynamic range, stability and tolerance. The digital filters are programmable. Multiplexing and adaptive filtering are possible.

Disadvantages of digital filters The bandwidth of the discrete signal is limited by the sampling frequency. The performance of the digital filter depends on the hardware used to

implement the filter.

16. Define ripples in a filter. The limits of the tolerance in the magnitude of passband and stopband are

called ripples. The tolerance in passband is denoted as δp and that in stopband is denoted as δs. 17. Classify the filters based on frequency response.

Based on frequency response, the filters can be classified into lowpass, highpass bandpass and bandstop filters.

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18. What are the requirements for an analog filters to be stable and causal? i. The analog filter transfer function H(s) should be a rational function of s

and the coefficients of s should be real. ii. The poles should lie on the left half of s-plane. iii. The number of zeros should be less than or equal to number of poles.

19. Distinguish between IIR and FIR filters. (DEC-2004)

The filter design starts from ideal frequency response. By taking inverse fourier transform of ideal frequency response, the desired impulse response is obtained, which consists of infinite number of samples.

The digital filter design by selecting only N samples of the impulse response are called FIR filters. The digital filters designed by considering all the infinite samples of impulse response are called IIR filters. 20. Compare IIR and FIR filters.

IIR Filter FIR Filter i. All the infinite samples of

impulse response are considered.

ii. The impulse response cannot be directly converted to digital filter transfer function.

iii. The design involves design of analog filter and then transforming analog filter to digital filter.

iv. The specifications include the desired characteristics for magnitude response only.

v. Linear phase characteristics cannot be achieved.

i. Only N samples of impulse response are considered.

ii. The impulse response can be directly converted to digital filter transfer function.

iii. The digital filter can be directly designed to achieve the desired specification.

iv. The specifications include the desired characteristics for both magnitude and phase response.

v. Linear phase filter can be easily designed.

21. How a digital IIR filter is designed? (DEC-2012)

For designing a digital IIR filter, first an equivalent analog filter is designed using any one of the approximation technique and the given specifications. The result of the analog filter design will be an analog filter transfer function H(s). The analog filter transfer function is transformed to digital filter transfer function H(z) using either Bilinear or impulse invariant transformation.

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22. Mention any two techniques for digitizing the transfer function of an analog filter.

The bilinear transformation and the impulse invariant transformation are the two techniques available for digitizing the analog filter transfer function. 23. What are the properties that are maintained same in the transformation of

analog to digital filer? The analog filter should be stable and causal for effective transformation to

digital filters. While transforming the analog filer to digital filters these two properties (i.e. stability and causality) are maintained same, which means that the transformed digital filer should also be stable and causal. 24. What is aliasing? (NOV-2010)

The phenomena of high frequency sinusoidal components acquiring the identity of low frequency sinusoidal components after sampling is called aliasing. The aliasing problem will arise if the sampling rate does not satisfy the Nyquist sampling criteria. 25. What is frequency warping?

In bilinear transformation the relation between analog and digital frequencies is non-linear. When the s-plane is mapped in to z-plane using bilinear transformation, this non-linear relationship introduce distortion in frequency axis, which called frequency warping. 26. What is butterworth approximation? (NOV-2009)

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. 27. How the poles of butterworth transfer function are located in s-plane?

The poles of the normalized butterworth transfer function symmetrically lies on a unit circle in s-plane with angular spacing of π/N.

28. What is the properties of butterworth filter?

i. The butterworth filters are pole design. ii. At the cutoff frequency Ωc, the magnitude of normalized butterworth filter

is 1/√2. iii. The filter order N, completely specifies the filter and as the value of N

increases the magnitude response approaches the ideal response.

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iv. The magnitude is maximally flat at the origin and monotonically decreasing with increasing Ω.

29. What is chebyshev approximation? (NOV-2008) In chebyshev approximation, the approximation function is selected such that

the error is minimized over a prescribed band of frequencies. 30. How does the order of the filter affect the frequency response of chebyshev

filter? From the magnitude response of type-I chebyshev filter it can be observed

that the magnitude response approaches the ideal response as the order of the filter is increased.

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UNIT - 3 FIR Filter Design

Syllabus: Structures of FIR – Linear phase FIR filter – Fourier series - Filter design

using windowing techniques (Rectangular Window, Hamming Window, Hanning Window), Frequency sampling techniques – Finite word length effects in digital Filters: Errors, Limit Cycle, Noise Power Spectrum. Two mark questions: 1. What are FIR filters? (DEC-2008)

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 filter 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. What are the techniques of designing FIR filters? (NOV-2004)

There are three well-known methods for designing FIR filters with linear phase. These are 1) windows method 2) Frequency sampling method 3) Optimal or mini-max design. 5. What is the reason that FIR filter is always stable?

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

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6. What are the properties of FIR filter? (NOV-2011) 1. FIR filter is always stable. 2. A realizable filter can always be obtained. 3. FIR filter has a linear phase response.

7. 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.

8. What are the advantages of FIR filters? (DEC-2007) Linear phase FIR filter can be easily designed. Efficient realization of FIR filter exist as both recursive and non-

recursive structures. FIR filters realized non-recursively are always stable. The round-off noise can be made small in non-recursive realization of

FIR filters. 9. What are the disadvantages of FIR filters?

The duration of impulse response should be large to realize sharp cutoff filters.

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

10. 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. 11. When cascade form realization is preferred in FIR filters?

The cascade form realization is preferred when complex zeros with absolute magnitude less than one.

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12. What are the conditions to be satisfied for constant phase delay in linear

phase FIR filters? (APR-2009) The conditions for constant phase delay are Phase delay, α = (N-1)/2 (i.e., phase delay is constant) Impulse response, h(n) = -h(N-1-n) (i.e., impulse response is

antisymmetric) 13. How constant group delay & phase delay is achieved in linear phase FIR

filters? The following conditions have to be satisfied to achieve constant group delay

& phase delay. 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 antisymmetric) 14. 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. Antisymmetric impulse response when N is odd. Antisymmetric impulse response when N is even.

15. 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

Fourier series method and window method Frequency sampling method. Optimal filter design methods.

16. What are the desirable characteristics of the frequency response of window

function? The desirable characteristics of the frequency response of window function are

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

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

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17. What is Gibb’s phenomenon (or Gibb’s Oscillation)? (NOV-2008) 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 passband and stopband. This effect is known as Gibb’s phenomenon (or Gibb’s Oscillation). 18. Write the procedure for designing FIR filter using frequency-sampling

method. (DEC-2008) Choose the desired (ideal) frequency response Hd(w). Take N-samples of Hd(w) to generate the sequence Take inverse DFT of to get the impulse response h(n). The transfer function H(z) of the filter is obtained by taking z-transform

of impulse response. 19. 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. 20. Write the characteristic features of rectangular window.

The main lobe width is equal to 4π/N. The maximum side lobe magnitude is –13dB. The side lobe magnitude does not decrease significantly with increasing w.

21. 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 passband and stopband. The attenuation in the stopband is constant and cannot be varied.

22. 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.

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23. How are phase distortion and delay distortion introduced? The phase distortion is introduce 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 a constant within the

desired frequency range. 24. Explain briefly the need for scaling in the digital filter realization.

Nov/Dec 2007 CSE To prevent overflow, the signal level at certain points in the digital filters must

be scaled so that no overflow occur in the adder. 25. What are the desirable and undesirable features of FIR Filters?

(May/June 2006)-ECE The width of the main lobe should be small and it should contain as much of

total energy as possible. The side lobes should decrease in energy rapidly as w tends to π.

26. Define backward and forward predictions in FIR lattice filter.

(NOV 2005 IT) The reflection coefficient in the lattice predictor is the negative of the cross

correlation coefficients between forward and backward prediction errors in the lattice.

27. Give the Kaiser Window function. (Apr/May 2004)-ECE

The Kaiser Window function is given by WK(n) = I0(β) / I0(α) , for |n| ≤ (M-1)/2

Where α is an independent variable determined by Kaiser. Β = α [ 1 – (2n/M-1)2]

28. Give the equation specifying Barlett and hamming window.

(NOV 2004 ITDSP) The transfer function of Barlett window wB(n) = 1-(2|n|)/(N-1), ((N-1)/2)≥n≥-((N-1)/2) The transfer function of hamming window whm(n) = 0.54+0.46cos((2πn)/(N-1), ((N-1)/2)≥n≥-((N-1)/2) α = 0.54

29. What is meant by FIR filter? And why is it stable? (APR 2004 ITDSP)

FIR filter Finite Impulse Response. The desired frequency response of a FIR filter can be represented as

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∞ Hd(ejω)= Σ hd(n)e-jωn

n= -∞ If h(n) is absolutely summable (i.e., Bounded Input Bounded Output Stable). So, it is in stable. 30. Give the equation specifying Barlett and hamming window.

(NOV 2004 ITDSP) The transfer function of Barlett window wB(n) = 1-(2|n|)/(N-1), ((N-1)/2)≥n≥-((N-1)/2) The transfer function of Hamming window whm(n) = 0.54+0.46cos((2πn)/(N-1), ((N-1)/2)≥n≥-((N-1)/2) α = 0.54

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UNIT - 4 FINITE WORDLENGTH EFFECTS

Syllabus: Fixed point and floating point number representations – ADC –Quantization-

Truncation and Rounding errors -Quantization noise – coefficient quantization error – Product quantization error - Overflow error – Round-off noise power - limit cycle oscillations due to product round off and overflow errors – Principle of scaling

Two mark questions: 1. What do finite word length effects mean? (DEC-2008)

The effects due to finite precision representation of numbers in a digital system are called finite word length effects. 2. List some of the finite word length effects in digital filters.

Errors due to quantization of input data. Errors due to quantization of filter co-efficient Errors due to rounding the product in multiplications Limit cycles due to product quantization and overflow in addition.

3. What are the different formats of fixed-point representation? (APR-2004)

Sign magnitude format One’s Complement format Two’s Complement format.

In all the three formats, the positive number is same but they differ only in representing negative numbers. 4. Explain the floating-point representation of binary number.

The floating-point number will have a mantissa part. In a given word size the bits allotted for mantissa and exponent are fixed. The mantissa is used to represent a binary fraction number and the exponent is a positive or negative binary integer. The value of the exponent can be adjusted to move the position of binary point in mantissa. Hence this representation is called floating point. 5. What are the types of arithmetic used in digital computers?

The floating point arithmetic and two’s complement arithmetic are the two types of arithmetic employed in digital systems.

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6. What is truncation? The truncation is the process of reducing the size of binary number by

discarding all bits less significant than the least significant bit that is retained. In truncation of a binary number of b bits all the less significant bits beyond bth bit are discarded. 7. What is rounding? (DEC-2009)

Rounding is the process of reducing the size of a binary number to finite word size of b-bits such that, the rounded b-bit number is closest to the original un-quantized number. 8. Explain the process of upward rounding?

In upward rounding of a number of b-bits, first the number is truncated to b-bits by retaining the most significant b-bits. If the bit next to the least significant bit that is retained is zero, then zero is added to the least significant bit of the truncated number. If the bit next to the least significant bit that is retained is one then one is added to the least significant bit of the truncated number. 9. What are the errors generated by A/D process? (APR-2008)

The A/D process generates two types of errors. They are quantization error and saturation error. The quantization error is due to representation of the sampled signal by a fixed number of digital levels. The saturation errors occur when the analog signal exceeds the dynamic range of A/D converter. 10. What is quantization step size?

In digital systems, the numbers are represented in binary. With b-bit binary we can generate 2b different binary codes. Any range of analog value to be represented in binary should be divided into 2b levels with equal increment. The 2b levels are called quantization levels and the increment in each level is called quantization step size. If R is the range of analog signal then, Quantization step size, q = R/2b 11. How the digital filter is affected by quantization of filter coefficients?

The quantization of the filter coefficients will modify the value of poles & zeros and so the location of poles and zeros will be shifted from the desired location. This will create deviations in the frequency response of the system. Hence the resultant filter will have a frequency response different from that of the filter with un-quantized coefficients.

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12. What is meant by product quantization error? (APR-2006) In digital computations, the output of multipliers i.e., the product are

quantized to finite word length in order to store them in registers and to be used in subsequent calculations. The error due to the quantization of the output of multiplier is referred to as product quantization error.

13. Why rounding is preferred for quantizing the product?

In digital system rounding due to the following desirable characteristic of rounding performs the product quantization

The rounding error is independent of the type of arithmetic The mean value of rounding error signal is zero. The variance of the rounding error signal is least.

14. What are limit cycles? (DEC-2004)

In recursive systems when the input is zero or some nonzero constant value, the nonlinearities die to finite precision arithmetic operations may cause periodic oscillations in the output. These oscillations are called limit cycles. 15. What is zero input limit cycles?

In recursive system, the product quantization may create periodic oscillations in the output. These oscillations are called limit cycles. If the system output enters a limit cycles, it will continue to remain in limit cycles even when the input is made zero. Hence these limit cycles are also called zero input limit cycles. 16. What is dead band? (APR-2008)

In a limit cycle the amplitudes of the output are confined to a range of values, which is called dead band of the filter. 17. Define noise transfer function (NTF)? (DEC-2006)

The Noise Transfer Function is defined as the transfer function from the noise source to the filter output. The NTF depends on the structure of the digital networks. 18. How the sensitivity of frequency response to quantization of filter

coefficients is minimized? The sensitivity of the filter frequency response to quantization of the filter

coefficients is minimized by realizing the filter having a large number of poles and zeros as an interconnection of second order sections. Hence the filter can be realized in cascade or parallel form, in which the basic buildings blocks are first order and second order sections.

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19. What are the two types of limit cycles? (NOV-2008) The two types of limit cycles are zero input limit cycles and overflow limit

cycles. 20. How the system output can be brought out of limit cycle?

The system output can be brought out of limit cycle by applying an input of Large magnitude, which is sufficient to drive the system out of limit cycle. 21. What is saturation arithmetic? (DEC-2008)

In saturation arithmetic when the result of the arithmetic operation exceeds the dynamic range of number system, then the result is set to maximum or minimum possible value. If the upper limit is exceeded then the result is set to maximum possible value. If the lower limit is exceeded then the result is set to minimum possible value. 22. What is overflow limit cycle?

In fixed point addition the overflow occurs when the sum exceeds the finite word length of the register used to store the sum. The overflow in addition may leads to oscillation in the output which is called overflow limit cycle. 23. How overflow limit cycles can be eliminated?

The over flow limit cycles can be eliminated either by using saturation arithmetic or by scaling the input signal to the adder. 24. What is the drawback in saturation arithmetic? (NOV-2006)

The saturation arithmetic introduces nonlinearity in the adder which creates signal distortion. 25. What are the two types of quantization employed in a digital system?

The two type of quantization in a digital system are truncation and rounding. 26. What is the range of error in rounding?

The rounding error is same in all the three types of fixed point representation. The range of rounding error is [-2-b/2] to [+2-b/2]. 27. Explain the fixed point representation of binary numbers.

In fixed point representation of binary number in a given word size, the bits allotted for integer part and fraction part of the numbers are fixed. Therefore the position of binary points is fixed. The most significant bit is used to represent the sign of the number.

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Binary point

Sign bit bi-bits for integer bf-bits for fraction

Fig. Fixed point representation of binary numbers. 28. How the input quantization noise is represented in LTI system?

The quantized input signal of a digital system can be represented as a sum of un-quantized signal x(n) and error signal e(n) as shown in below fig.

e(n) xq(n)

x(n) y(n) Fig. Representation of input quantization noise in an LTI system 29. What is mean by coefficient inaccuracy? (APR-2014)

In digital computation the filter coefficients are represented in binary. With b-bit (excluding sign bit) binary we can generate only 2b different binary numbers and they are called quantization levels. Any filter coefficients has to be fitted into any one of the quantization levels. Hence the filter coefficients are quantized to represent in binary and the quantization introduces errors in filter coefficients. Therefore the coefficients cannot be accurately represented in a digital system and this problem is referred to as coefficient inaccuracy. 30. What are the assumptions made regarding the statistical independence of

the various noise sources in the digital filter? The assumption made regarding the statistical independence of the noise

source are, i. Any two different samples from the same noise source uncorrelated. ii. Any two different noise source, when considered as random processes are

uncorrelated. iii. Each noise source is uncorrelated with the input sequence.

h(n)

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UNIT - 5 DSP APPLICATIONS

Syllabus: Multi-rate signal processing: Decimation, Interpolation, Sampling rate

conversion by a rational factor – Adaptive Filters: Introduction, Applications of adaptive filtering to equalization.

Two mark questions: 1. What is multi-rate signal processing? (APR-2012)

The theory of processing signals at different sampling rates is called multi-rate signal processing. 2. Define down sampling. (or Decimation)

Down sampling a sequence x(n) by a factor M is the process of picking every Mth sample and discarding the rest. 3. What is mean by up-sampling? (or interpolation)

Up-sampling by a factor L is the process of inserting L-1 zeros between two consecutive samples. 4. If the spectrum of sequence x(n) is X(ejw), then what is the spectrum of a

signal down-sampled by factor 2? (DEC-2013) Y(ejw)=(1/2)[X(ejw/2)+ X(ejw((w/2)-π)]

5. If the Z-transform of a sequence x(n) is X(z) then what is the Z-transform of

a sequence down-sampled by a factor M? Y(z)= (1/M) ∑ 푋(z(1/M)e(-j2πk/M))

6. If the z-transform of a sequence x(n) is X(z) then what is the z-transform of

a sequence up-sampled by a factor L? Y(z)= X(zL)

7. What is the need for anti-imaging filter after up-sampling a signal?

The frequency spectrum of up-sampled signal with a factor L, contains (L-1) additional images of the input spectrum. Since we are not interested in image spectra, a low-pass filter with a cutoff frequency wc = (π/L) can be used after up-sampler. This filter is known as anti-imaging filter.

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8. What is the need for anti-aliasing filter prior to down-sampling? The spectra obtained after down-sampling a signal by a factor M is the sum

of all uniformly shifted and stretched version of original spectrum scaled by a factor (1/M). If the original spectrum is not band limited to (π/M), then down-sampling will cause aliasing. In order to avoid aliasing the signal x(n) is to be band limited to ± (π/M). This can be done by filtering the signal x(n) with a low pass filter with a cutoff frequency of (π/M). This filter is known as anti-aliasing filter. 9. Define Sampling rate conversion.

Sampling rate conversion is a process of converting a signal from a given rate to a different rate. Sampling rate conversion by a rational factor (L/M) can be achieved by first performing interpolation by the factor L and then performing decimation by the factor M. 10. What is multirate DSP system? (APR-2014)

The discrete time system that employs sampling rate conversion while processing the discrete time signal is called multirate DSP system. 11. What are the various basic methods of sampling rate conversion in digital

domain? The basic methods of sampling rate conversion are decimation (or

downsampling) and interpolation (or upsampling). 12. Give any two applications of multirate DSP system.

1. Sub-band coding of speech signals and image compression. 2. Oversampling A/D and D/A converters for high quality audio systems and

digital storage systems. 13. Write some advantages of multirate processing.

1. The reduction in number of computation. 2. The reduction in memory requirement. 3. The reduction in finite word length effects.

14. What is anti-aliasing filter?

The low pass filter used at the input of decimator is called anti-aliasing filter.it is used to limit the bandwidth of an input signal to (π/D) in order to prevent the aliasing of output spectrum of decimator for decimation by D.

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15. What is an anti-imaging filter? (DEC-2014) The low pass filter used at the output of an interpolator is called anti-imaging

filter.it is used to eliminate the multiple images in the output spectrum of the interpolator. 16. Write a short note on sampling rate conversion by a rational factor.

When sampling rate conversion is required by a non-integer factor, then sampling rate conversion is performed by the rational factor [I/D]. In this method, the signal is first interpolated by an integer factor I, then passed through a low pass filter with bandwidth minimum of [(π/I), (π/D)], and finally decimated by an integer factor, D. 17. What is poly-phase decomposition?

The process of dividing a filter into a number of sub-filter which differ only in phase characteristics is called poly-phase decomposition. 18. Write a short note on multistage implementation of sampling rate

conversion. When the sampling rate conversion factor I or D is very large then the

multistage sampling rate conversion will be computationally efficient relalization. In multistage interpolation, the interpolation by I is realized as cascade of

interpolators with sampling rate multiplication factors I1,I2,……IL, where I= I1x I2 x ……x IL.

In multistage decimation, the decimation by D is realized as a cascade of decimator with sampling rate reduction factors D1, D2,….DL, where D= D1 x D2 x…..x DL. 19. Draw the structure of anti-aliasing filter. (APR-2004)

The structure of anti-aliasing filter is x(n) anti-aliasing filter down sampler y(n)

v(n)

20. Draw the frequency domain representation of downsampler. x(n) y(n) = x(Dn) x(ejw) Y(ejw)=[1/D] x(ejw/D)

h(n) D

D

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21. Write the expression for output spectrum, Y(ejw) of an interpolator in terms of input spectrum, X(ejw).

Output spectrum, Y(ejw) = X(ejwI) Where, I = integer sampling rate multiplication factor of interpolator.

22. Write the expression for output spectrum, Y(ejw) of decimator in terms of

input spectrum, X(ejw).

Output spectrum, Y(ejw) = [1/D] 푋(푒 ( )/ ) Where, D= integer sampling rate reduction factor of decimator.

23. Draw the Multirate signal processing system with analysis and synthesis

filter banks. (APR-2009)

24. Draw the structure of Brickwall filters.

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25. Draw the Characteristics of real filters in subband decomposition.

26. Draw the Time-frequency resolutions of a signal of length N = 8 with the

three-stage filter bank.

27. Draw the structure of M-fold decimator. (DEC-2010) X(n) yD(n) yD(n)= x(Mn)

For an input sequence x(n), select only the samples which occur at integer multiples of M. The other samples are thrown away.

Aliasing will occur in yD(n) unless x(n) is sufficiently bandlimited loss of information. 28. Draw the structure of L-folder expander.

For an input sequence x(n), insert L − 1 zeros between each sample. x(n) can always be recovered from yE(n) no loss of information, no aliasing.

M

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X(n) yE(n) YE(n)= x(Mn) 29. Develop an expression for the output y(n) as a function of the input x(n)

for the multirate structure of below fig. (NOV-2010)

30. Determinr the computational complexity of a single stage decimator

designed to reduce the sampling rate from 60kHz to 3kHz. The decimation filter is to be designed as in equiripple FIR filter with a passband edge at 1.25kHz, a passband ripple of 0.02, and a stopband ripple of 0.01. use the total multiplications per second as a measure of the computational complexity. (DEC-2012)

Answer:

L

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Dsp_ece_5th Sem (2mark Q&A)