1. Define discrete time signal.

A discrete time signal x (n) is a function of an independent variable that is an integer. a discrete time signal is not defined at instant between two successive samples.

2. Define discrete time system.

A discrete or an algorithm that performs some prescribed operation on a discrete time signal is called discrete time system.

3. What are the elementary discrete time signals?

• Unit sample sequence (unit impulse)

δ (n)= {1 n=0

0 Otherwise

• Unit step signal

U (n) ={ 1 n>=0

0 Otherwise

• Unit ramp signal

Ur(n)={n for n>=0

0 Otherwise

• Exponential signal

x (n)=an where a is real

x(n)-Real signal

4. State the classification of discrete time signals.

The types of discrete time signals are

• Energy and power signals

• Periodic and Aperiodic signals

• Symmetric(even) and Antisymmetric (odd) signals

5. Define energy and power signal.

∞

E=∑│x (n) │2

n=-∞

If E is finite i.e. 0<E<∞, then x (n) is called energy signal.

If P is finite in the expression P=Lt (1/2N+1) EN, the signal is called a power signal.

N->∞

6. Define periodic and aperiodic signal.

A signal x (n) is periodic in period N, if x (n+N) =x (n) for all n. If a signal does not satisfy this equation, the signal is called aperiodic signal.

7. Define symmetric and antisymmetric signal.

A real value signal x (n) is called symmetric (even) if x (-n) =x (n). On the other hand the signal is called antisymmetric (odd) if x (-n) =x (n).

8. State the classification of systems.

• Static and dynamic system.

• Time invariant and time variant system.

• Causal and anticausal system.

• Linear and Non-linear system.

• Stable and Unstable system.

9. Define dynamic and static system.

A discrete time system is called static or memory less if its output at any instant n depends almost on the input sample at the same time but not on past and future samples of the input.

e.g. y(n) =a x (n)

In anyother case the system is said to be dynamic and to have memory.

e.g. (n) =x (n)+3 x(n-1)

10.Define time variant and time invariant system.

A system is called time invariant if its output , input characteristics dos not change with time.

e.g.y(n)=x(n)+x(n-1)

A system is called time variant if its input, output characteristics changes with time.

e.g.y(n)=x(-n).

11.Define linear and non-linear < href="http://studentwebsite.blogspot.com/2009/09/software-engineering-model-question.html" target="_blank">system.

Linear system is one which satisfies superposition principle.

Superposition principle:

The response of a system to a weighted sum of signals be equal to the corresponding weighted sum of responses of system to each of individual input signal.

i.e., T [a1x1(n)+a2x2(n)]=a1T[x1(n)]+a2 T[x2(n)]

e.g.y(n)=nx(n)

A system which does not satisfy superposition principle is known as non-linear system.

e.g.(n)=x2(n)

12.Define causal and anticausal system.

The system is said to be causal if the output of the system at any time ‘n’ depends only on present and past inputs but does not depend on the future inputs.

e.g.:- y (n) =x (n)-x (n-1)

A system is said to be non-causal if a system does not satisfy the above definition.

13.Define stable and unable system.

A system is said to be stable if we get bounded output for bounded input.

14.What are the steps involved in calculating convolution sum?

The steps involved in calculating sum are

• Folding

• Shifting

• Multiplication

• Summation

15. Define causal LTI system.

The LTI system is said to be causal if

h(n)=0 for n<0

16. Define stable LTI system.

The LTI system is said to be stable if its impulse response is absolutely summable. ∞

i.e. │y(n)│ =∑ │h(k)│ < ∞

k= - ∞

17.what are the properties of convolution sum

The properties of convolution sum are

• Commutative property.

• Associative law.

• Distributive law.

18.State associative law

The associative law can be expressed as

[x(n)*h1(n)]*h2(n)=x(n)[h1(n)*h2(n)]

Where x(n)-input

h1(n)-impulse response.

19.State commutative law

The commutative law can be expressed as

x(n)*h(n)=h(n)*x(n)

20. State distributive law

The distributive law can be expressed as

x(n)*[h1(n)+h2(n)]=x(n)*h1(n)+x(n)*h2(n)

21.Define Z-transform

Z- transform can be defined as

∞

X(Z)=∑ x(n)z-n

n=-∞

22.Define Region of convergence

The region of convergence (ROC) of X(Z) the set of all values of Z for which X(Z) attain final value.

23.State properties of ROC.

• The ROC does not contain any poles.

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

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

• If X(Z) is anticasual,then ROC includes Z=0.

24.state the properties of Z-transform.

i) linearity:-

Z Z

if x1(n)↔X1(Z) and x2(n)↔X2(Z)

then Z

a1x1(n)+a2x2(n)↔a1X1(Z)+a2X2(Z)

ii)Time shifting

Z

if x(n)↔X(Z)

then Z

x(n-k)↔Z-KX(Z)

iii)Scaling in Z-domain

Z

if x(n)↔X(Z)

Z

then anx(n)↔X(a-1Z)

iv)Time reversal

Z

if x(n)↔ X(Z)

Z

then x(-n)↔ X(Z-1)

v)Differtiation in Z domain

Z

nx(n)↔-Zdz X(Z)

vi)convolution of two sequences

Z Z

if x1(n)↔X1(Z) and x2(n)↔x2(Z)

Z

then x1(n)*x2(n)↔X(Z)=X1(Z).X2(Z)

vii)correlation

Z Z

if x1(n)↔X1(Z) and x2(n)↔X2(Z)

then ∞ Z

rx1x2(l=∑x1(n) x2(nl)↔Rx1x2(Z)=X1(Z) .X2(Z-1)

n=-∞

25.State the methods for evaluating inverse Z-transform.

• Direct valuation by contour integration.

• Expansion into series of terms in the variable Z and Z-1.

• Partial fraction expansion and look up table.

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

DFT(Analysis Equation)

N-1 nk

X(k)= ∑ x(n) WN , WN = e-j2∏/N

n=0

IDFT(Synthesis Equation)

N-1 - nk

x(n)= 1/N ∑ X(k) WN , WN = e-j2∏/N

k=0

27.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

28.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

N-1

x3(n)=∑ x1(m)x2((n-m))N

m=0

29.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.

30.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 zero padding are

1)We can get better display of the frequency spectrum.

2)With zero padding the DFT can be used in linear filtering.

31.Define sectional 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 up into smaller sections. These sections are processed separately one at a time and controlled later to get the output.

32.What are the two methods used for the sectional convolution?

The two methods used for the sectional convolution are

1)the overlap-add method and 2)overlap-save method.

33.What is 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 cicular convolution of xi(n) and 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 blocks.This method is called overlap-add method.

34.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.

35.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.

36.What is FFT?

The Fast Fourier Transform is an algorithm used to compute the DFT. It makes use of the symmetry and periodicity properties of twiddle factor to effectively reduce the DFT computation time.It is based on the fundamental principle of decomposing the computation of DFT of a sequence of length N into successively smaller DFTs.

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

The number of multiplications and additions required to compute N point DFT using radix-2 FFT are N log2 N and N/2 log2 N respectively,.

38.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.

39.What is DIT algorithm?

Decimation-In-Time algorithm is used to calculate the DFT of a N point sequence. The idea is to break the N point sequence into two sequences, the DFTs of which can be combined to give the DFt of the original N point sequence.This algorithm is called DIT because the sequence x(n) is often splitted into smaller sub- sequences.

40.What DIF algorithm?

It is a popular form of the FFT algorithm. In this the output sequence X(k) is divided into smaller and smaller sub-sequences , that is why the name Decimation In Frequency.

41.Draw the basic butterfly diagram of DIT algorithm.

The basic butterfly diagram for DIT algorithm is

Where a and b are inputs and A and B are the outputs.

42.Draw the basic butterfly diagram of DIF algorithm.

The basic butterfly diagram for DIF algorithm is

Where a and b are inputs and A and B are outputs.

43.What are the applications of FFT algorithm?

The applications of FFT algorithm includes

1) Linear filtering

2) Correlation

3) Spectrum analysis

44.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.

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

Linear convolution

If x(n) is a sequence of L number of samples and h(n) with M number of samples, after convolution y(n) will have N=L+M-1 samples.

It can be used to find the response of a linear filter.

Zero padding is not necessary to find the response of a linear filter.

Circular convolution

If x(n) is a sequence of L number of samples and h(n) with M samples, after convolution y(n) will have N=max(L,M) samples.

It cannot be used to find the response of a filter.

Zero padding is necessary to find the response of a filter.

46.What are differences between overlap-save and overlap-add methods.

Overlap-save method

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

Each data block consists of the last M-1 data points of the previous data block followed by L new data points

In each output block M-1 points are corrupted due to aliasing as circular convolution is employed

To form the output sequence the first

M-1 data points are discarded in each output block and the remaining data are fitted together

Overlap-add method

In this method the size of the input data block is L

Each data block is L points and we append M-1 zeros to compute N point DFT

In this no corruption due to aliasing as linear convolution is performed using circular convolution

To form the output sequence the last

M-1 points from each output block is added to the first M-1 points of the succeeding block

47.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.

48. 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.

49. 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

51.Distinguish between FIR filters and IIR filters.

FIR filter

These filters can be easily designed to have perfectly linear phase.

FIR filters can be realized recursively and non-recursively.

Greater flexibility to control the shape of their magnitude response.

Errors due to round off noise are less severe in FIR filters, mainly because feedback is not used.

IIR filter

These filters do not have linear phase.

IIR filters are easily realized recursively.

Less flexibility, usually limited to specific kind of filters.

The round off noise in IIR filters is more.

52. 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.

53.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.

54. List the steps involved in the design of FIR filters using windows.

1.For the desired frequency response Hd(w), find the impulse response hd(n) using Equation

π

hd(n)=1/2π∫ Hd(w)ejwndw

-π

2.Multiply the infinite impulse response with a chosen window sequence w(n) of length N to obtain filter coefficients h(n),i.e.,

h(n)= hd(n)w(n) for |n|≤(N-1)/2

= 0 otherwise

3.Find the transfer function of the realizable filter

(N-1)/2

H(z)=z-(N-1)/2 [h(0)+∑ h(n)(zn+z-n)]

n=0

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

The desirable characteristics of the window are

1.The central lobe of the frequency response of the window should contain most of the energy and should be narrow.

2.The highest side lobe level of the frequency response should be small.

3.The side lobes of the frequency response should decrease in energy rapidly as ω tends to п .

56.Give the equations specifying the following windows.

a. Rectangular window

b. Hamming window

c. Hanning window

d. Bartlett window

e. Kaiser window

a. Rectangular window:

The equation for Rectangular window is given by

W(n)= 1 0 ≤ n ≤ M-1

0 otherwise

b. Hamming window:

The equation for Hamming window is given by

WH(n)= 0.54-0.46 cos 2пn/M-1 0 ≤ n ≤ M-1

0 otherwise

c. Hanning window:

The equation for Hanning window is given by

WHn(n)= 0.5[1- cos 2пn/M-1 ] 0 ≤ n ≤ M-1

0 otherwise

d. Bartlett window:

The equation for Bartlett window is given by

WT(n)= 1-2|n-(M-1)/2| 0 ≤ n ≤ M-1

M-1

0 otherwise

e. Kaiser window:

The equation for Kaiser window is given by

Wk(n)= Io[α√1-( 2n/N-1)2] for |n| ≤ N-1

Io(α) 2

0 otherwise

where α is an independent parameter.

57. What is the necessary and sufficient condition for linear phase characteristic in FIR filter?

The necessary and sufficient condition for linear phase characteristic in FIR filter is, the impulse response h(n) of the system should have the symmetry property i.e.,

H(n) = h(N-1-n)

where N is the duration of the sequence.

58.What are the advantages of Kaiser window?

o It provides flexibility for the designer to select the side lobe level and N

o 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

59. 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.

60. 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.

61. Draw the direct form realization of FIR system.

DIRECT REALIZATION FIR SYSME Diagram

62. Draw the direct form realization of a linear Phase FIR system for N even.

LINEAR PHASE FIR SYSTEM FOR N EVEN DIAGRAM

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

LINEAR PHASE FIR SYSTEM N ODD DIAGRAM

64. 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.

64. Draw the M stage lattice filter.

M STAGE LATTICE FILTER DIAGRAM

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

Filter coefficient.

αm(0) = 1

αm(m) = km

αm(k) = αm-1(k) + αm(m) • αm-1(m-k)

66. State the equations used to convert the FIR filter coefficients to the lattice filter

Coefficient.

For an M_stage filter , αm-1(0) =1 and km = αm(m)

αm-1(k) = αm(k) - αm(m) • αm(m-k) , 1≤k≤m-1

1-αm2 (m)

67. 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

68. State the advantage of direct form ΙΙ structure over direct form Ι structure.

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

69. 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.

70. Mention the procedures for digitizing the transfer function of an analog filter.

The two important procedures for digitizing the transfer function of an analog filter are

• Impulse invariance method.

• Bilinear transformation method.

71. 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.

72. 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 Z-plane.

• The right half of S-plane maps into the region outside the circle of radius ½ in the Z-plane.

• The j Ω-axis maps onto the perimeter of the circle of radius ½ in the Z-plane.

73. 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,

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

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

75. What is bilinear transformation?

The bilinear transformation is a mapping that transforms the left half of S-plane into the unit circle in the Z-plane only once, thus avoiding aliasing of frequency components.

The mapping from the S-plane to the Z-plane is in bilinear transformation is

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

76. 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 j Ω-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.

77. Write a short note on pre-warping.

The effect of the non-linear compression at high frequencies can be compensated. When the desired magnitude response is piece-wise constant over frequency, this compression can be compensated by introducing a suitable pre-scaling, or pre-warping the critical frequencies by using the formula.

78. What are the advantages & disadvantages of bilinear transformation?

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.

79. What is the advantage of cascade realization?

Quantization errors can be minimized if we realize an LTI system in cascade form.

80. Define signal flow graph.

A signal flow graph is a graphical representation of the relationships between the variables of a set of linear difference equations.

81. What is transposition theorem & transposed structure?

The transpose of a structure is defined by the following operations.

• Reverse the directions of all branches in the signal flow graph

• Interchange the input and outputs.

• Reverse the roles of all nodes in the flow graph.

• Summing points become branching points.

• Branching points become summing points.

According to transposition theorem if we reverse the directions of all branch transmittance and interchange the input and output in the flowgraph, the system function remains unchanged.

82.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.

83.What is meant by fixed point number?.

In fixed point number the position of a binary point is fixed. The bit to the right represent the fractional part and those to the left is integer part.

84.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

85. 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.

86. 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.

87. 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.

88. 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.

89 What are the advantages of floating pint representation?

1.Large dynamic range 2.overflow is unlikely.

90.What are the quantization errors due to finite word length registers in digital filters?

1.Input quantization errors2.Coefficient quantization errors3.Product quantization errors

91.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.

92. .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.

93. What is input quantization error?.

The input quantization errors arise due to A/D conversion.

94.What are the different quantization methods?

Truncation and Rounding

95.What is truncation?

Truncation is a process of discarding all bits less significant than LSB that is retained

96. 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.

97.What are the two types of limit cycle behavior of DSP?.

1.Zero limit cycle behavior 2.Over flow limit cycle behavior

98.What are the methods to prevent overflow?

1.Saturation arithmetic and2.Scaling

99.State some applications of DSP?

Speech processing ,Image processing, Radar signal processing.

100.Draw the Block diagram of channel vocoder.