UNIT I

1

1. If H(k) is the N-point DFT of a sequence h(n), Prove that H(k) and H(N-K) are complex

conjugates. (Nov2008)

If DFT[x(n)]=X(k)

Then DFT[x*(n)]=X*(N-k)=X*((-k))N

Proof:

N 1

j2 kn

N 1 j2 kn

*

DFT [x* (n )] =x* (n ) e N = x(n )e N

n =0 n =0

N 1 j2 n (N k )

*

=X * (N k )

= x(n )e N

n =0

DFT[x*(N-n)] = X*(k) Proof:

IDFT [X * (k )] = N1

1 N 1

= X (k ) e

N k =0

N 1 j2 kn

X * (k ) e N

k =0

j2 k (N n )

*

N

1

N 1 j2 kn

*

= X (k ) e N

N k =0

= x*(N-n)

Therefore DFT[x*(N-n)] = X*(k)

2. What are the differences and similarities between DIF and DIT algorithms? (Nov2008)

Sl.No DIT FFT DIF FFT

1 The time domain sequence is The DFT x(k) is decimated decimated

2 Input sequence is to be given in bit The DFT at the output is in bit reversed

reversed order. order.

3 First calculate 2-point DFTs and Decimates the sequence step by step to

combines them 2-point sequence and calculate DFT.

4 Suitable for calculating inverse Suitable for calculating DFT. DFT.

3. Define the properties of convolution. (April 2008, Nov 2005)

Commutative Law 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) 4. Draw the basic butterfly diagram of radix-2 FFT. (April 2005, May 2007 & April 2008)

a A=a+Wnb

b W n

B =a – Wn b

-1

5. State and prove parseval’s relation for DFT. (Nov 2007)

If DFT[x(n)] = X(k)

and DFT[y(n)] = Y(k)

Then N 1 N 1

x(n )y* (n ) = 1 X(k )Y* (k )

n =0 N

k =0

Proof:

N 1 N 1

1 N 1

x(n )y* (n ) = x(n ) Y(k

n =0 n =0 N k =0

N 1

N 1

= 1 x(n ) Y* (k ) e

N n =0 k =0

N 1

= 1 X(k )Y* (k )

N

k =0

j2 kn

*

) e N

j2 kn

N 1 j2 kn

1 N 1

N = Y* (k ) x(n ) e N

N k =0 n =0

Hence proved 6. What do you mean by the term “bit reversal” as applied to FFT

In DIT algorithm we can find that for the output sequence to be in a natural order (i.e., X(k) , k=0,1,2,….N-1) the input sequence has to be stored in a shuffled order. For an 8-point DIT algorithm the input sequence is in the order x(0), x(4), x(2),x(6),x(1),x(5),x(3) and x(7). We can see that when N is a power of 2 , the input sequence must be stored in bit-reversal order for the output to be computed in a natural order. For N = 8 the bit-reversal process is shown in table.

Input sample Binary Bit reversed Bit reversed sample

index representation binary index

0 000 000 0

1 001 100 4

2 010 010 2

3 011 110 6

4 100 001 1

5 101 101 5

6 110 011 3

7 111 111 7

7. What are the advantages of FFT algorithm over direct computation of DFT? (May 2007)

The complex multiplication in the FFT algorithm is reduced by (N/2) log2N times. Processing speed is very high compared to the direct computation of DFT.

8. The first five DFT coefficients of a sequence x(n) are x(0) = 20, x(1) = 5+j2, x(2) = 0, x(3)=0.2+j0.4, X(4) = 0.

Determine the remaining DFT coefficients. (May 2007)

By complex conjugate property x(5)=0.2-j0.4,x(6)=0,x(7)=5-j2 9. Define symmetric and Anti symmetric signals. How do you prevent aliasing while sampling a CT signal? (May

2007) A real valued signal x(n) is called

symmetric if X (n) = X (-n)

On the other hand, a signal x(n) is called antisymmetric

X (-n) = -X (n) 10. what is the necessary and sufficient condition on the impulse response for stability? (May 2007)

The necessary and sufficient condition for the impulse response is given by +∞

∑ |h (n)|< ∞

n=-∞ 11. Define Complex Conjugate of DFT property. (May 2007) DFT

If x(n)↔X(k) then

N

X*(n)↔(X*(-k))N = X*(N-K)

12. What is FFT? (Nov 2006)

The fast Fourier transform is an algorithm is used to calculate the DFT. It is based on fundamental principal of

decomposing the computation of DFT of a sequence of the length N in to successively smaller discrete Fourier Transforms. The FFT algorithm provides speed increase factor when compared with direct computation of the DFT.

13. State sampling theorem? (Nov 2006)

Sampling is the process to convert analog time domain continuous signal into discrete time

domain signal. But it is the process of converting only time domain not in amplitude domain.

Nyquist criteria:

We sample the signal based on the following condition i.e., fs ≥2fm

Where fx = Sampling frequency

Fm = maximum signal frequency

If these above conditions are not satisfied we will meet the following demerits after the sampling process. 1. Guard band , 2. Aliasing Effect

14. What is BIBO Stability? What is necessary and sufficient condition for BIBO stability?

(May 2006 , Nov 2004)

Any system is said to be BIBO stable of and only if every bounded input gives a bounded output. The BIBO stability depends on the impulse response of the system. The necessary and sufficient condition

for BIBO stability 15. How will you perform linear convolution via circular convolution? (May 2006)

Let the length of x(n) be L, length of h(n) be M. then linear convolution of x(n) and h(n) can be obtained

through following steps.

i. Append x(n) with M-1 zeros. Hence its length will be L+M-1 ii. Append h(n) with L-1 zeros. Hence its length will be L+M-1

iii. Perform circular convolution of above sequences. The result is linear convolution of length L+M-1

16. How many multiplications and additions are required to compute N-point DFT using radix-2 FFT?

In computing N-point DFT by this method the number of stages of computation will be m-times. The

number ‘r’ is called the radix of the FFT algorithms.

In radix-2-FFT, the total number of complex additions are reduced to N log2N and total number of

complex multiplications are reduced too (N/@)log2N. 17. What is decimation-in-time algorithm?

The computation of 8-point DFT using radix -2 DIT – FFT, involves three stages of computation. Here N= =

23 therefore r=2 and m=3.

The given 8 point sequence is decimated to 2 point sequences. For each 2 point sequence, the 2 point DFT

is computed. From the result of 2-point DFT the 4-point DFT can be computed. From the result of 4-point DFT,

the 8-point DFT can be computed.

Let the given sequence be X(o),X(1),X(2), X(3), X(4), X(5), X(6), X(7) which consists of 8 samples. 18. What is decimation-in-frequency algorithm?

In decimation in frequency algorithm the frequency domain sequence X(k) is decimated. In this method,

the output DFT sequence X(k) is divided into smaller sequence.

19. Derive the necessary and sufficient condition for an LTI system to be BIBO stable. (April 2005) A system is BIBO stable, if for every bounded input, the output is finite. Mathematically if

|x(t)|∞ < ∞

And

||y(t)|| < ∞ then the system is stable.

The necessary and sufficient condition for Continuous time signal is stable if and only if

-∞

|| h(t) ||1 = ∫|h(n)| dt < ∞

∞

In discrete time system

∞

||h||1 = ∑ |h(n)|

n=-∞

20. Define DFT pair? (April 2004 & May 2007)

The DTFT pairs are

X(k) = x(n)e-j2πkn/N

x(n) = X(k)ej2πkn/N

21. What is aliasing? (Nov 2003)

If we operate the sampler at fx < fm, the frequency components of the frequency spectrum will overlap

with each other i.e., the lower frequency of the second frequency component will overlap with higher

frequency of the first frequency component. This overlapping effect is called as Aliasing effect. For avoiding

overlapping of high and low frequency components, we have to use low-pass filter to cut the unwanted high

frequency components. 22. Give any two properties of DFT

a)Periodicity x(k+n)=x(k)

b)Linearity DFTa1x1(n)+a2x2(n)=a1x1(k)+a2x2(k). 23. Explain Linearity property of DFT

DFTx (n)=x (k)&DFTy (n)=Y (k) For any real valued constant a &b . DFTa x (n)+b y (n)=a X (k)+a Y (k)

24. What are the applications of FFT algorithms? (May/June 2009, April/May 2008)

The applications of FFT algorithm include

1. Linear filtering

2. Correlation

3. Spectrum analysis 25. What are twiddle factors of the DFT? (N/D 2007, M/J 2006)

The complex valued phase factor WN is called as twiddle factor,which is an nth root of unity as WN = e-j2π/N . 26. How many additions and multiplications are needed to compute N-point FFT? (N/D 2007)

The total number of complex additions = Nlog2N

The total number of complex multiplication = N/2 log2N. 27. Calculate the number of multiplications in 64 point DFT using FFT? (M/J 2007, 2009)

Number of complex multiplications is given by = N/2 log2N

Here N = 64

= 64/2 log264 = 32 log226 = (32 x 6)log22 = 192

Therefore Number of multiplications = 192. 28. Find the values of WNk when N=8 and k=2 and also for k=3. (M/J 2007)

We know that WN = e-j2π/N Here N=8, W8 = e-j2π/8 = e-jπ/4

When k=2, W82 = (e-jπ/4)2 = e-jπ/2 = cos(π/2) – jsin(π/2) = -j

When k=3, W83 = (e-jπ/4)3 = cos(3π/4) – jsin(3π/4) = (-1/√2) – j(1/√2) 29. Determine the circular convolution of the sequence x1(n) = 1,2,3,1 and x2(n) = 4,3,2,1. (N/D 2007)

1 1 3 2

4

4 + 3 + 6 + 2

15

2 1 1 3 3 =

8 + 3 + 2 + 3 =

16

3 2 1 1 2 12 + 6 + 2 + 1 21

1 3 2 1 1 4 + 9 + 4 + 1 18

X3(n) = 15,16,21,18 30. Find the linear convolution of 1,0,1 and 2,0,2. (N/D 2007)

Y(n) = 2,0,4,0,2

31. What is zero padding ? What are its uses? Let the sequence x(n) has a length L. If we want to find the N-point DFT (N>L) of the sequence x(n), we have

to add (N-L) zeros to the sequence x(n). This is known as zero padding. The uses of padding a sequence with

zeros are (a) We can get “better display” of the frequency spectrum.

(b) With zero padding, the DFT can be used in linear filtering.

32. Distinguish between linear and circular convolution of two sequences.

Linear convolution Circular convolution

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

and h(n) with M number of samples, after h(n) with M number of samples, after convolution y(n) will contain N= L+M-1 convolution y(n) will contain N= Max(L,M) samples. samples. 2. Linear convolution can be used to find the Circular convolution cannot be used to find the response of a linear filter. response of a linear filter.

3. Zero padding is not necessary to find the Zero padding is necessary to find the response of

response of a linear filter. a linear filter. 33. What is meant by sectioned convolution?

If the data sequence x(n) is of long 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. 34. What are the two methods used for the sectioned convolution?

The two methods used for the sectioned convolution are (1) the overlap-add method and (2) overlap-save method.

35. Write briefly about overlap-add method?

In this method the size of the input data block xi(n) is L. To each data block we append M-1 zeros and perform

N-point (N = L+ M-1) 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. 36. State the difference between (i) overlap-save method (ii) overlap-add method. Overlap-save method Overlap-add method

1 In this method the size of the input data block is N=L+M-1

In this method the size of the input

data block is L.

2 Each data block consists of the last M-1 data Each data block is L points and we

points of the previous data block followed by L new data points.

append M-1 zeros to compute N-point DFT.

3 In each output block M-1 points are corrupted

In this no corruption due to aliasing, as

due to aliasing, as circular convolution is linear convolution is performed using

employed. circular convolution.

4 To form the output sequence the first M-1 To form the output sequence, the last data points are discarded in each output

block M-1 points from each output block is

and the remaining data are fitted together. added to the first (m-1) points of the

succeeding block.

37. What are the steps involved in calculating convolution sum? The steps involved in calculating sum are

· Folding · Shifting · Multiplication · Summation

38. How to obtain the output sequence of linear convolution through circular convolution?

Consider two finite duration sequences x(n) and h(n) of duration L samples and M samples. The linear convolution of these two sequences produces an output sequence of duration L+M-1 samples, whereas, the circular convolution of x(n) and h(n) give N samples where N=max(L,M).In order to obtain the number of samples in circular convolution equal to L+M-1, both x(n) and h(n) must be appended with appropriate number of zero valued samples. In other words by increasing the length of the sequences x (n) and h(n) to L+M-1 points and then circularly convolving the resulting sequences we obtain the same result as that of linear convolution. 39. Define circular convolution.

Let x1(n) and x2(n) are finite duration sequences both of length N with DFTs X1(K) and X2(k) If X3(k)=X1(k)X2(k) then the sequence x3(n) can be obtained by circular convolution defined as

40. Why FFT is needed?

The direct evaluation DFT requires N2 complex multiplications and N2 –N complex additions.Thus for large values of N direct evaluation of the DFT is difficult.By using FFT algorithm the number of complex computations can be reduced. So we use FFT. 41. Why the computations in FFT algorithm is said to be in place?

Once the butterfly operation is performed on a pair of complex numbers (a,b) to produce (A,B), there is no need to save the input pair. We can store the result (A,B) in the same locations as (a,b). Since the same storage locations are used troughout the computation we say that the computations are done in place. 42. What are the differences and similarities between DIF and DIT algorithms? Differences:

1)The input is bit reversed while the output is in natural order for DIT, whereas for DIF the output is bit reversed while the input is in natural order.

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. Both algorithms can be done in place and both need to perform bit reversal at some place during the computation.

43. What is meant by radix-2 FFT?

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 algorithm.

44. What is overlap-save method?

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.In circular convolution of xi(n) with h(n) the first M-1 points will not agree with the linear convolution of xi(n) and h(n) because of aliasing, the remaining points will agree with linear convolution. Hence we discard the first (M-1) points of filtered section xi(n) N h(n). This process is repeated for all sections and the filtered sections are abutted together. 45. State the properties of DFT.

1) Periodicity 2) Linearity and symmetry 3) Multiplication of two DFTs 4) Circular convolution 5) Time reversal 6) Circular time shift and frequency shift 7) Complex conjugate 8)Circular correlation

46. Define DFT and IDFT (or) What are the analysis and synthesis equations of DFT?

DFT(Analysis Equation) N-1 nk

IDFT(Synthesis Equation)

UNIT – II 1. Show that the filter with h (n) = [-1, 0, 1] is a linear phase filter. (Nov 2008, May 2007)

From the above equation we can find θ(ω) = -ω which is the proportional to ω. Hence the filter h(n) is a linear Phase filter.

2. What are the merits and demerits of FIR filters? (Nov 2005 & April 2008)

FIR filters that have ideal linear phase characteristics can be easily designed. FIR filters realized non-recursively are always stable.

Errors arising from quantization of signals and finite word length effects are usually less critical for FIR filter designs as these realization do not have feedback FIR filters are implemented through FFT algorithms, which greatly reduced its processing time. 3. In the design of FIR digital filters, how is Kaiser window different from other windows? (Nov 2007)

It provides flexibility for the designer to select the side lobe level and N. It has the attractive property that the side lobe level can be varied continuously from the low value in the Blackman window to the high value in the rectangular window. 4. State the condition for a digital filter to be causal and stable. (May 2007)

The response of the causal system to an input does not depend on future values of that input, but depends only on the present and/or past values of the input. A filter is said to be stable, bounded-input bounded output stable, if every bounded input produces a bounded output. A bounded signal has amplitude that remains finite. 5. What is the condition satisfied by linear phase FIR filter? (Nov/Dec 2003 & May 2007)

Linear phase is of the form ∟θ (ω) = k ω

Here k is constant. Thus phase shift is linearly proportional to frequency. For linear phase, the impulse response should satisfy following condition.

h (n) = ± h (M-1-n) 6. Give any two properties of Butterworth filter and chebyshev filter. (Nov/Dec 2006, May/June 2006, Apr 2005 & Nov 2004)

a. The magnitude response of the Butterworth filter decreases monotonically as the frequency increases (Ώ) from 0 to ∞.

b. The magnitude response of the Butterworth filter closely approximates the ideal response as the order N increases.

c. The poles on the Butterworth filter lies on the circle. d. The magnitude response of the chebyshev type-I filter exhibits ripple in the pass band. e. The poles of the Chebyshev type-I filter lies on an ellipse.

7. What are the desirable and undesirable features of FIR Filters? (May2006)

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

8. Define Hanning and Blackman window functions. (May 2006)

11 The window function of a causal hanning window is given by

WHann(n) = 0.5 – 0.5cos2πn/ (M-1), 0≤n≤M-1

0, Otherwise

The window function of non-causal Hanning window I s expressed by

WHann(n) = 0.5 + 0.5cos2πn/ (M-1), 0≤|n|≤(M-1)/2

0, Otherwise

The width of the main lobe is approximately 8π/M and the peak of the first side lobe is at -32dB.

The window function of a causal Blackman window is expressed by

WB(n) = 0.42 – 0.5 cos2πn/ (M-1) +0.08 cos4πn/(M-1), 0≤n≤M-1

= 0, otherwise The window function of a non causal

Blackman window is expressed by

WB(n) = 0.42 + 0.5 cos2πn/ (M-1) +0.08 cos4πn/(M-1), 0≤|n|≤(M-1)/2

= 0, otherwise

The width of the main lobe is approximately 12π/M and the peak of the first side lobe is at -58dB. 9. Write the magnitude function of Butterworth filter. What is the effect of varying order of N on magnitude and

phase response? (Nov 2005) |H(jΏ)|2 = 1 / [ 1 + (Ώ/ΏC)2N] where N= 1,2,3,….

10. Mention the necessary and sufficient condition for linear phase characteristics in FIR filter. (Nov 2005)

The necessary and sufficient conditions is that the phase function should be linear function w, which in turn requires constant phase delay (or) constant phase and group delay i.e., Q(w) α w

Q(w) = - α w -π≤w≤ π 11. What is linear phase? What is the condition to be satisfied by the impulse response in order to have a linear phase? (Apr 2005 & Nov 2003)

For a filter to have linear phase the phase response θ(w) α w is the angular frequency.

The linear phase filter does not alter the shape of the signal. The necessary and sufficient condition for a filter to have linear phase.

h(n) = ± h(N-1-n); 0 ≤ n ≤ N-1 12. List the characteristics of FIR filters designed using window functions. (Nov 2004)

the Fourier transform of the window function W(ejw) should have a small width of main lobe containing as much of the total energy as possible

the fourier transform of the window function W(ejw) should have side lobes that decrease in energy rapidly as w to π. Some of the most frequently used window functions are described in the following sections.

13. Give the Kaiser Window function. (Apr 2004)

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] 14. 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 sample and previous input samples.

15. What are the different types of filters based on frequency response?

Based on frequency response the filters can be classified as 1. Lowpass filter 2. Highpass filter 3. Bandpass filter 4. Bandreject filter

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

Advantages: 1. FIR filters have exact linear phase. 2. FIR filters are always stable. 3. FIR filters can be realized in both recursive and non recursive structure. 4. Filters with any arbitrary magnitude response can be tackled using FIR

sequence. Disadvantages: 1. For the same filter specifications the order of FIR filter design can be as high as 5 to 10

times that in an IIR design. 2. Large storage requirement is requirement 3. Powerful computational facilities required for the implementation.

17. What are the design techniques of designing FIR filters?

There are three well known methods for designing FIR filters with linear phase .They are (1.)Window method (2.)Frequency sampling method (3.)Optimal or minimax design.

18. What is Gibb’s phenomenon?

One possible way of finding an FIR filter that approximates H(ejw) would be to truncate the infinite Fourier series at n=±(N-1/2).Direct truncation of the series will lead to fixed percentage overshoots and undershoots before and after an approximated discontinuity in the frequency response.

19. What are the desirable characteristics of the window function?

13 20. List the steps involved in the design of FIR filters using windows. 21. What are the advantages of Kaiser Window?

It provides flexibility for the designer to select the side lobe level and N It has the attractive property that the side lobe level can be varied continuously from

the low value in the Blackman window to the high value in the rectangular window 22. What is the principle of designing FIR filter using frequency sampling method?

In frequency sampling method the desired magnitude response is sampled and a linear phase response is specified .The samples of desired frequency response are identified as DFT coefficients. The filter coefficients are then determined as the IDFT of this set of samples.

23. For what type of filters frequency sampling method is suitable?

Frequency sampling method is attractive for narrow band frequency selective filters where only a few of the samples of the frequency response are non zero.

24. Draw the direct form realization of FIR system.

14 25. When cascade form realization is preferred in FIR filters?

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

26. State the equations used to convert the lattice filter coefficients to direct form

FIR Filter coefficient. 27. Draw the direct form realization of a linear Phase FIR system for N even.

28. Draw the direct form realization of a linear Phase FIR system for N odd 15

29. Draw the M stage lattice filter. 30. State the equations used to convert the FIR filter coefficients to the lattice filter Coefficient.

31. What are the properties of FIR filter?

1. FIR filter is always stable.

2. A realizable filter can always be obtained.

3. FIR filter has a linear phase response. 32. What do you understand by linear phase response? For a linear phase filter θ(ω)α ω , the linear filter does not alter the shape of the original signal. If the phase response of the filter is nonlinear the output signal may be distorted one. In many cases a linear phase characteristic is required throughout the passband of the filter to preserve the shape of a given signal within the passband. An IIR

filter cannot produce a linear phase. The FIR filter can give linear phase, when the impulse response of the filter is symmetric about its mid-point.

33. What are the disadvantages of Fourier series method?

In designing FIR filter using Fourier series method the infinite duration impulse response is truncated at

n = ±(N-1/2). Direct truncation of the series will lead to fixed percentage overshoots and undershoots before

and after an approximated discontinuity in the frequency response. 34. What is the need for employing window technic for FIR filter design? OR What is window and why it is

necessary?

One possible way of finding an FIR filter that approximates H(ejω) would be to truncate the infinite Fourier

series at n = ±(N-1/2). Abrupt truncation of the series will lead to oscillations in the passband and stopband.

These oscillations can be reduced through the use of less abrupt truncation of the Fourier series. This can be

achieved by multiplying the infinite impulse response by a finite weighing sequence w(h), called a window. 35. Define Rectangular and Hamming window functions. 36. Compare FIR and IIR filters. (May 2007) Sl.No IIR FIR

1 H(n) is infinite duration H(n) is finite duration

2 Poles as well as zeros are present. Sometimes These are all zero filters.

pole filters are also designed.

3 These filters use feedback from output. They These filters do not use feedback. They

recursive filters. nonrecursive.

4 Nonlinear phase response. Linear phase Linear phase response for h(n) = ± h(m-1-n)

obtained if H(z) = ±Z-1H(Z-1)

5 These filters are to be designed for stability These are inherently stable filters

6 Number of multiplication requirement is less. More

7 More complexity of implementation Less complexity of implementation

UNIT – III

1. What is prewarping? (Nov 2003,2008)

When bilinear transformation is applied, the discrete time frequency is related continuous time frequency as,

Ω = 2tan-1ΩT/2

This equation shows that frequency relationship is highly nonlinear. It is also called frequency warping. This effect

can be nullified by applying prewarping. The specifications of equivalent analog filter are obtained by following

relationship,

Ω = 2/T tan ω/2

This is called prewarping relationship. 2. What is the relation betweeen analog and digital frequency in impulse invariant transformation?(April 2008)

ΩT= ω

3. State the condition for a digital filter to be causal and stable. (May 2007)

The response of the causal system to an input does not depend on future values of that input, but depends

only on the present and/or past values of the input.

A filter is said to be stable, bounded-input bounded output stable, if every bounded input produces a

bounded output. A bounded signal has amplitude that remains finite. 4. Find the digital transfer function H (z) by using impulse invariant method for the analog transfer function H(s) = 1/(S+2). Assume T=0.5sec.

H(s) = 1/(s+2) The system function of the digital filter is obtained by

H (z) = 1/ (1-e-2Tz-1)

Since T=o.5 sec

H (z) = 1/ (1-.067Z-1)

5. Mention any two procedures for digitizing the transfer function of an analog filter. (Nov 2006)

1. Impulse Invariant Technique 2. Bilinear Transform Technique

6. what are the parameters that can be obtained from the chebyshev filter specification? (Nov 2006 May

2007)

(or) 18 Give the equation for the order N, major, minor and axis of an ellipse in case of chebyshev filter. (Nov 2005)

N ≥ cosh-1 (λ/ε) / cosh-1(ΏS/ ΏP)

Where λ = √ (100.1αs – 1)

ε = √ (100.1αp – 1)

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

(May 2006)

Advantages:

· The bilinear transformation provides one-to-one mapping. · Stable continuous systems can be mapped into realizable, stable digital systems. · There is no aliasing.

Disadvantage:

· The mapping is highly non-linear producing frequency, compression at high frequencies. · Neither the impulse response nor the phase response of the analog filter is preserved in a

digital filter obtained by bilinear transformation. 8. What is impulse invariant mapping? What is its limitation? (Apr/May 2005)

The philosophy of this technique is to transform an analog prototype filter into an IIR discrete time filter

whose impulse response [h(n)] is a sampled version of the analog filter’s impulse response, multiplied by T.

This procedure involves choosing the response of the digital filter as an equi-spaced sampled version of the

analog filter. 9. What is frequency warping? (Nov2004 & May 2007)

The bilinear transform is a method of compressing the infinite, straight analog frequency axis to a finite one

long enough to wrap around the unit circle only once. This is also sometimes called frequency warping. This

introduces a distortion in the frequency. This is undone by pre-warping the critical frequencies of the analog filter

(cutoff frequency, center frequency) such that when the analog filter is transformed into the digital filter, the

designed digital filter will meet the desired specifications. 10. What are the limitations of impulse invariant mapping technique? (Apr2004)

The impulse invariance technique is appropriate only for band limited filter like low pass filter. Impulse

invariance design for high pass or band stop continuous-time filters, require additional band limiting to avoid

severe aliasing distortion, if impulse designed is used. Thus this method is not preferred in the design of IIR filters

other than low-pass filters. 11. Give the transform relation for converting low pass to band pass in digital domain. (Apr 2004)

Low pass with cut – off frequency ΏC to band –pass with lower cut-off frequency Ώ1 and higher cut-off frequency Ώ2:

S ------------- ΏC ( s2 + Ώ1 Ώ2) / s (Ώ2 - Ώ1)

The system function of the high pass filter is then

H(s) = Hp ΏC ( s2 + Ώ1 Ώ2) / s (Ώ2 - Ώ1)

12. Give the bilinear transformation. (Nov2003)

The bilinear transformation method overcomes the effect of aliasing that is caused due to the analog

frequency response containing components at or beyond the nyquist frequency. The bilinear transform is a

method of compressing the infinite, straight analog frequency axis to a finite one long enough to wrap around

the unit circle only once.

S = (2/T) (Z-1) (Z+1)

13. State the structure of IIR filter?

IIR filters are of recursive type whereby the present o/p sample depends on present i/p, past i/p samples and o/p samples. The design of IIR filter is realizable and stable. The impulse response h(n) for a realizable filter is h(n)=0 for n ≤ 0

14. State the advantage of direct form II structure over direct form I structure.

In direct form II structure, the number of memory locations required is less than that of direct form I structure.

15. How one can design digital filters from analog filters?

· Map the desired digital filter specifications into those for an equivalent analog filter. · Derive the analog transfer function for the analog prototype. · Transform the transfer function of the analog prototype into an equivalent digital

filter transfer function. 16. What do you understand by backward difference?

One of the simplest method for converting an analog filter into a digital filter is to approximate the differential equation by an equivalent difference equation.

d/dt y(t)=y(nT)-y(nT-T)/T The above equation is called backward difference equation.

17. What is the mapping procedure between S-plane & Z-plane in the method of

mapping differentials? What are its characteristics?

The mapping procedure between S-plane & Z-plane in the method of mapping of differentials is given by H(Z) =H(S)|S=(1-Z-1)/T

The above mapping has the following characteristics · The left half of S-plane maps inside a circle of radius ½ centered at Z= ½ in the Zplane. · The right half of S-plane maps into the region outside the circle of radius ½ in the Z-plane. · The jΩ-axis maps onto the perimeter of the circle of radius ½ in the Z-plane.

18. What is meant by impulse invariant method of designing IIR filter?

In this method of digitizing an analog filter, the impulse response of resulting digital filter is a sampled version of the impulse response of the analog filter. The transfer function of analog filter in partial fraction form.

19. Give the bilinear transform equation between S-plane & Z-plane.

S=2/T(1-Z-1/1+Z-1) 20. What are the properties of bilinear transformation?

· The mapping for the bilinear transformation is a one-to-one mapping that is for every point Z, there is exactly one corresponding point S, and vice-versa.

· The jΩ-axis maps on to the unit circle |z|=1,the left half of the s-plane maps to the interior of the unit circle |z|=1 and the half of the s-plane maps on to the exterior of the unit circle |z|=1.

21. Define an IIR filter.

The filter 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. 22. Distinguish between IIR and FIR filters.

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 filters designed by selecting only N samples of the impulse response are called FIR filters. The digital filters designed by selecting all the infinite samples of impulse response are called IIR filters. 23. What are the requirements for an analog filter to be stable and causal?

(i) The analog filter transfer function Ha(s) should be a rational function os ‘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. 24. Write a brief note on the design of IIR filter. (OR) How a digital IIR filter is designed?

For designing a digital IIR filter, first an equivalent analog filter is designed using any one of the approximation technique and given specifications. The result of the analog filter design will

be an analog filter transfer function Ha(s). The analog filter transfer function is transformed to digital filter transfer function H(z) using either Bilinear or Impulse invariant transformation. 25. Mention the important features of IIR filters.

(i) The physically realizable IIR filters does not have linear phase. (ii) The IIR filter specifications includes the desired characteristics for the magnitude response only.

26. What are the advantages and disadvantages of digital

filters? Advantages of digital filters:

(i) High thermal stability due to absence of resistors, inductors and capacitors. (ii) The performance characteristics like accuracy, dynamic range, stability and

tolerance can be enhanced by increasing the length of the registers. (iii) The digital filters are programmable. (iv) Multiplexing and adaptive filtering are possible.

Disadvantages of digital filters:

(i) The bandwidth of the discrete signal is limited by the sampling frequency.

(ii) The performance of the digital filter depends on the hardware used to implement the filter. 27. What is the main objective of impulse invariant transformation?

The objective of this method is to develop an IIR filter transfer function whose impulse response is the sampled version of the impulse response of the analog filter. Therefore the frequency response characteristic of the analog filter is preserved. 28. What is the importance of poles in filter design?

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. 29. What is aliasing?

The phenomena of high frequency sinusoidal components acquiring the identity of low frequency sinusoidal components after sampling is called aliasing.i.e. aliasing is higher frequencies impersonating lower frequencies. The aliasing problem will arise if the sampling rate does not satisfy the Nyquist sampling criteria. 30. What is aliasing problem in impulse invariant method of designing digital filters?Why it is absent in bilinear transformation?

In impulse invariant mapping, the analog frequencies in the interval(2k-1)π/T≤Ω≤(2k+1)π/T (where k is an integer) maps into corresponding values of digital frequencies in the interval -π≤Ω≤π. Hence the mapping of Ω to ω is many-to-one.

This will result in high frequency components acquiring the identity of the low frequency components if the analog filter is not bandlimited. This effect is called aliasing. The aliasing can be avoided in bandlimited filters by choosing very small values of sampling time(or very high sampling frequency). The bilinear mapping is one-to-one mapping and so there is no effect of aliasing.

31. 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. 32. Compare the impulse invariant and bilinear transformations.

Impulse invariant transformation Bilinear transformation (i) It is many-to-one mapping. (i) It is one-to-one mapping. (ii) The relation between analog and (ii) The relation between analog and digital digital frequency is linear. frequency is nonlinear.

(iii) To prevent the problem of aliasing (iii) There is no problem of aliasing and so the

the analog filters should be bandlimited. analog filter need not be bandlimited. (iv) The magnitude and phase response (iv) Due to the effect of warping, the phase

of analog filter can be preserved by response of analog filter cannot be preserved.

choosing low sampling time or high But the magnitude response can be preserved

sampling frequency. by prewarping. 33. Write the properties of butterworth filter.

(i) The butterworth filters are all pole designs. (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. (iv) The magnitude is maximally flat at the origin and monotonocally decreasing with increasing Ω.

34. What is Chebyshev approximation?

In chebyshev approximation, the approximation function is selected such that the error is minimized over a prescribed band of frequencies. 35. What is type-I chebyshev approximation?

In type-I chebyshev approximation, the error function is selected such that, the magnitude response is equiripple in the passband and monotonic in the stopband. 36. What is type-II chebyshev approximation?

In type-I chebyshev approximation, the error function is selected such that, the magnitude response is monotonic in the passband and equiripple in the stopband. The type-II magnitude response is called inverse Chebyshev response. 37. How the order of the filter affects 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. 38. How the poles of chebyshev transfer function are located in s-plane?

The poles of the chebyshev transfer function symmetrically lies on an ellipse in s-plane. 39. Write the properties of chebyshev type-I filters.

(i) The magnitude response is equiripple in the passband and monotonic in the stopband. (ii) The chebyshev type-I filters are all pole designs.

(iii) The normalized magnitude function has a value of 1/√(1+ε2) at the cutoff frequency Ωc. (iv) The magnitude response approaches the ideal response as the value of N increases.

40.Compare the Butterworth and chebyshev Type-I filters.

Butterworth Chebyshev Type-I (i) All pole design. (i) All pole design. (ii) The poles lie on a circle in s-plane. (ii) The poles lie on an ellipse in s-plane. (iii) The magnitude response is maximally

(iii) The magnitude response is equiripple in

flat at the origin and monotonically passband and monotonically decreasing in

decreasing function of Ω. the stopband.

(iv) The normalized magnitude response (iv) The normalized magnitude response has a

has a value of 1/√2 at the cutoff frequency

value of 1/√(1+ε2) at the cutoff frequency

Ωc. Ωc.

UNIT - IV 1. Express the fraction (-9/32) in sign magnitude, 2’s complement notations using 6 bits.

(Nov 2008)

Sign magnitude : 1.01001

2’s complement : 1.10111

2. What are the three types of quantization error occurred in digital systems? ( Nov 2006 & Apr 2008)

Input quantization error Coefficient quantization error Product quantization error

3. What is meant by limit cycle oscillations? ((May 2006,Apr 2005 May 2007, Nov 2007 & Apr 2008)

In fixed point addition, overflow occurs due to excess of results bit, which are stored at the registers.

Due to this overflow, oscillation will occur in the system. Thus oscillation is called as an overflow limit cycle

oscillation. 5. Express the fraction(-7/32) in signed magnitude and two’s complement notations using 6 bits. (Nov

2007)

Sign magnitude : 1.00111 2’s complement : 1.11001 6. Express the fraction 7/8 and -7/8 in sign magnitude, 2’s complement and 1’s complement. (Nov 2006)

7/8 -7/8

Sign magnitude : 0.111 1.111

1’s complement : 0.000 1.000

2’s complement : 0.001

1.001

7. Define Sampling rate conversion. (May 2007)

Sampling rate conversion is the process of converting a signal from one sampling rate to another,

while changing the information carried by the signal as little as possible.

Sample rate conversion needed because different systems use different sampling rates. 8. Convert the number 0.21 into equivalent 6-bit fixed point number. (May 2007)

0.001101 9.Why rounding is preferred to truncation in realizing digital filter?(May2007)

Error introduced due to rounding operation is less compared to truncation. Similarly

quantization error due to rounding is independent of arithmetic operation. And mean of rounding

error is zero. Hence rounding is preferred over truncation in realizing digital filter.

10. What are the different quantization methods? (Nov 2006)

Amplitude quantization Vector quantization Scalar quantization

11. What is zero padding? Does zero padding improve the frequency resolution in the spectral estimate?

(Nov 2006)

The process of lengthening a sequence by adding zero—valued samples is called appending with zeros or zero padding.

Large dynamic range

Occurrence of overflow is very rare Higher accuracy

13. Give the expression for signal to quantization noise ratio and calculate the improvement

with an increase of 2 bits to the existing bit.(Nov2006,Nov2005)

SNRA / D = 16.81+6.02b-20log10 (RFS /σx) dB.

With b= 2 bits increase, the signal to noise ratio will increase by 6.02 X 2 = 12dB.

14. Draw the probability density function for rounding. (Nov 2005)

Shows the probability density function of error in rounding operation.

15. What is dead band?

(Nov 2004)

In a limit cycle the amplitude of the output are confined to a range of value, which is called dead band.

16. How can overflow limit cycles be eliminated? (Nov 2004)

Saturation Arithmetic Scaling

18. What is zero input limit cycle oscillation? (Apr 2004)

Zero Input Limit Cycles:

Zero input limit cycles are usually of lower amplitude in comparison with overflow limit

cycles. If the system enters to the limit cycles oscillations, it will continue even after input attains zero range.

19. What is steady state noise power at the output of an LTI system due to the quantization at the

input to L bits? (Nov 2003 &Apr 2004)

The steady state noise power is basically the variance of output noise. Π

σP = σe2.1/2π∫ |H(ω)|2 dw -π

12. List the advantages of floating point arithmetic. (Nov 2006)

Here σe2 is the variance of input error signal.

Σe2 = 2-2LRFS2 /48

π

σv2 = 2-2LRFS2 /48 X ½π ∫ |H

(ω)|2 dw - π

This equation gives steady state noise power due to quantization. 20. What is meant by finite word length effects in digital filters? (Nov 2003)

The digital implementation of the filter has finite accuracy. When numbers are represented in

digital form, errors are introduced due to their finite accuracy. These errors generate finite precision

effects or finite word length effects.

When multiplication or addition is performed in digital filter, the result is to be represented

by finite word length (bits). Therefore the result is quantized so that it can be represented by finite

word register. This quantization error can create noise or oscillations in the output. These effects are

called finite word length effects. 21. What is round-off noise error?

Rounding operation is performed only on magnitude of the number. Hence round-off noise

error is independent of type of fixed point representation. If the number is represented by bu bits

before quantization and b bits after quantization, then maximum round-off error will be (2_b-2-bu)/2. It is symmetric about zero.

22. What is meant by fixed point arithmetic? Give example?

In the fixed point arithmetic, the digits to the left of the decimal point represent the integer part of the number and digits to the right of the decimal point represent fractional part of the number. For example,

(1458.568)10 (1101.101)2 are the fixed point numbers note that base of the number system is also

written outside the bracket.

23. what are the different types of arithmetic in digital systems?

There are three types of arithmetic used in digital systems. They are fixed point arithmetic, floating point ,block floating point arithmetic.

24. What are the different types of fixed point arithmetic?

Depending on the negative numbers are represented there are three forms of fixed point arithmetic. They are sign magnitude,1’s complement,2’s complement

25. What is meant by sign magnitude representation?

For sign magnitude representation the leading binary digit is used to represent the sign. If it is equal to 1 the number is negative, otherwise it is positive.

26. What is meant by 1’s complement form?

In 1,s complement form the positive number is represented as in the sign magnitude form. To obtain the negative of the positive number ,complement all the bits of the positive number.

27. What is meant by 2’s complement form?

In 2’s complement form the positive number is represented as in the sign magnitude form. To obtain the negative of the positive number ,complement all the bits of the positive number and add 1 to the LSB.

28. What is meant by floating pint representation?

In floating point form the positive number is represented as F =2CM,where is mantissa, is a fraction such that1/2<M<1and C the exponent can be either positive or negative.

29. What are the advantages of floating pint representation?

1.Large dynamic range 2.overflow is unlikely.

30. 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.

31.What is product quantization error?

The product quantization errors arise at the out put 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 produces the error.

32.What are the different quantization methods?

Truncation and Rounding 33. What is overflow oscillations?

The addition of two fixed point arithmetic numbers cause overflow when the sum exceeds the word size available to store the sum. This overflow caused by adder make the filter output to oscillate between maximum amplitude limits. Such limit cycles have been referred to as overflow oscillations. 34. What are the two kinds of limit cycle behavior in DSP?

(i) Zero limit cycle oscillations (ii) Overflow limit cycle oscillations (iii)

35. What is meant by quantization step size?

Let us assume a sinusoidal signal varying between +1 and -1 having a dynamic range 2. If ADC used to convert the sinusoidal signal employs b+1 bits including sign bit, the number levels available for quantizing

x(n) is 2b+1. Thus the interval between successive levels. q= 2/(2b+1) = 2-b . where q is known as quantization step size. 36.Explain briefly the need for scaling in the digital filter implementation.

To prevent overflow, the signal level at certain points in the digital filters must be scaled so that no overflow occurs in the adder. 37. Why the limit cycle problem does not exist when FIR digital filter is realized in direct form or cascade form?

In the case of FIR filters, there are no limit cycle oscillations, if the filter is realized in direct form or cascade form since these structures have no feedback.

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

(i) The quantization error due to rounding is independent of the type arithmetic. (ii) The mean of rounding error is zero. (iii) The variance of the rounding error signal is low.

39. What is truncation?

Truncation is a process of discarding all bits less significant than LSB that is retained 40. What is Rounding?

Rounding a number to b bits is accomplished by choosing a rounded result as the b bit number closest number being unrounded.

41. State some applications of DSP?

Speech processing ,Image processing, Radar signal processing. 42. what is meant by A/D conversion noise?

A DSP contains a device, A/D converter that operates on the analog input x(t) to produce xq(t) which is binary sequence of 0s and 1s. At first the signal x(t) is sampled at regular intervals to produce a sequence x(n) is of infinite precision. Each sample x(n) is expressed in terms of a finite number of bits given the sequence xq(n). The difference signal e(n)=xq(n)-x(n) is called A/D conversion noise.

43.what is the effect of quantization on pole location?

Quantization of coefficients in digital filters lead to slight changes in their value. This change in value of filter coefficients modify the pole-zero locations. Sometimes the pole locations will be changed in such a way that the system may drive into instability.

44.which realization is less sensitive to the process of quantization?

Cascade form.

UNIT V