UNIT I : CLASSIFICATION OF SIGNALS

1. Difference between DSP and ASP.ASP Input signal given to the system is analog. Ex R,C,L, OP-AMP etc. DSP Input signal given to the system is digital. Ex Digital Computer, Digital Logic Circuits etc.

a. Compact and light in weight. b. More accurate i.e less sensitive to environment changes and noisec. Flexible, programmable and easily up-gradabled. Easy and lasting storage capacitye. Less cost.

2. Explain the block diagram of Digital system.

Analog Analog Signal signal

Most of the signals generated are analog in nature. Hence these signals are converted to digital form by the ADC. The DSP performs signal processing operations like filtering, multiplication, transformation or amplification etc operations over these digital signals. The digital output signal from the DSP is given to the DAC to generate analog signal again.

3. What are single channel - multi-channel signalsSingle channel signal signal is generated from single sensor or source. Ex. Speech or voice signal.Multi-channel signal signals are generated from multiple sensors or multiple sources Ex ECG signals.

Continuous time signals defined at any time instanceDiscrete time signals defined only at sampling instances.

Continuous values signal signal amplitude takes on all possible values on a finite or infinite rangeDiscrete values signal. signal takes values from a finite set of possible values.

Analog signals Continuous time & continuous amplitude signalsDigital signals Discrete time & discrete amplitude signals.

Deterministic signal value can be evaluated at any time without certainty.Random signal value can not be evaluated at any instant of time.

Periodic signal If x(n+N)= x(n) for all n where N is the fundamental period of the signal. Else non-periodic signals.

Symmetrical(Even) if x(n) = x(-n) Anti-symmetrical(Odd) x(-n) = -x(n)

Energy Signal Summation of magnitude squared values of x(n). The signal is called as energy signal if its energy if finite. A signal is called power signal if its power is finite. Ex: Energy of unit sample function is 1.

4. What is maximum range of discrete time frequencies & continuous time frequencies. Discrete time frequencies = -1/2 to 1/2 cycles/sample or -∏ to +∏ rad/samplesContinuous time frequencies = - ∞ to +∞

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ADC DIGITAL SYSTEM

DAC

5. Prove that discrete time signals are periodic only if frequency is rational. What is the condition for periodicity of DT signal.

6. CT periodic signals are converted into DT signals by sampling. DT signal may not be periodic. Explain this statement with suitable example. X(t) = sin(10) t CT signal (Periodic) Fs=1 t=nTS

X(n)= sin(10) n DT signal (Non-periodic). This is because Discrete frequency is not rational.

7. The highest rate of oscillation is achieved when the discrete frequency is –∏. Explain this statement with suitable example.

8. Prove that any discrete time signal is represented as a combination of even and odd signals with an example.Even component of signal =[ x(n) + x(-n) ] / 2 Odd component of signal =[ x(n) - x(-n) ] / 2Example: X(n)={1,2,1} Xe(n)={0.5,1,1,1,0.5} Xo(n)={-0.5,-1,0,1,0.5}

9. Explain the importance of unit sample signal. Unit sample is given as input to the system H. Output of the system will be h(n)

called as unit impulse response. Once we know the unit impulse response , we can find out the output of the same system for all type of inputs. (Linear Convolution).

10. What are different test signals used In DSP. Unit ramp, unit step and unit sample are three most used test signals in DSP.

Exponential and sine ways can also be used in DSP.

11. Which statement is correct? ∞ ∑ x (k) h(n – k ) (1) k= -∞ ∞ ∑ x (k) δ(n – k ) (2) k= -∞

12. What are Static or dynamic systems. Static Output depends on input sample at same time. Dynamic Output also depends upon past or future samples of input.

TIV If its IO characteristic does not change with shift of time.

Linear If it satisfies superposition theoremLet x1(n) and x2(n) are two input sequences, then the system is said to be linear if and only if T[a1x1(n) + a2x2(n)]=a1T[x1(n)]+a2T[x2(n)] (Superposition Theorem)

Causal If output of system depends only past and present inputs samples. Non-causal If output of system also depends on future inputs.

Stable If every bounded input produces a bounded output. Unstable If any bounded input produces an unbounded output.

13. How the discrete time signal is represented as weighted impulses.Let X(n) = {2,5,2}. The signal x(n) can also be written as X(n)= x(-1) δ(n+1) + x(0) δ(n) + x(1) δ(n-1).

14. Explain linear convolution. Will it applicable for Non-linear systems or Time variant systems.

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In linear convolution we decompose input signal into sum of elementary signal. Now the elementary input signals are taken into account and individually given to the system. Now using linearity property Whatever output response we get for decomposed input signal, we simply add it & this will provide us total response of the system to any given input signal. Linear Convolution states that

y(n) = x(n) * h(n)  ∞

y(n) =   ∑ x (k) h(n – k ) k= -∞

15. What are various properties of linear 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)

16. Explain when LSI system is causal. LSI system is causal if and only if h(n) =0 for n<0.

17. Explain when LSI system is stable.LSI system is stable if its unit sample response is absolutely summable.

∞   ∑ |h(k)| < ∞k=-∞

18. How the LSI system is represented by constant coefficient difference equation. (Generalized Difference equation)Difference equation of the generalized LSI system is given as N My(n)=-∑ ak y(n–k)+∑ bk x(n–k) k=1 k=0

19. What is sampling process. Why it is necessary.It is the process of converting continuous time signal into a discrete time signal by

taking samples of the continuous time signal at discrete time instants.

20. What is sampling theorem. What is Nyquist rate. Sampling Theorem states that if the highest frequency in an analog signal is Fmax

and the signal is sampled at the rate fs > 2Fmax then x(t) can be exactly recovered from its sample values. This sampling rate is called Nyquist rate of sampling.

If sampling frequency is less than Nyquist rate, then it is called under sampling. Under sampling creates aliasing. In aliasing high frequencies appear as low frequencies. 21. What is aliasing. Explain with example. How to avoid aliasing. Example:Case 1: X1(t) = cos 2∏ (10) t Fs= 40 Hz i.e t= n/Fs x1[n]= cos 2∏(n/4)= cos (∏/2)n

Case 2: X1(t) = cos 2∏ (50) t Fs= 40 Hz i.e t= n/Fs x1[n]= cos 2∏(5n/4)=cos 2∏(1+ ¼)n

=cos (∏/2)nThus the frequency 50 Hz, 90 Hz , 130 Hz … are alias of the frequency 10 Hz at the sampling rate of 40 samples/sec. To avoid aliasing sampling frequency should be selected as per sampling theorem and pass the signal through pre-alias filter before sampling.

22. What is quantization & coding.The process of converting a discrete time continuous amplitude signal into a digital

signal by expressing each sample value as a finite number of digits is called quantization.

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In the encoding operation, the quantization sample value is converted to the binary equivalent of that quantization level. If 16 quantization levels are present, 4 bits are required. Thus bits required in the coder is the smallest integer greater than or equal to Log2 L.

23. What is anti-aliasing filter. In which applications it is mostly used. The sampling rate of 6khz can be used for speech processing because speech

frequency range is up to 3kHz. But the speech signal also contains some frequency components more than 3khz. Hence a sampling rate of 6khz will introduce aliasing. Hence signal should be band limited to avoid aliasing. Thus the signal can be band limited by passing it through a filter (LPF) which blocks or attenuates all the frequency components outside the specific bandwidth.

24. Discuss Quantization NoiseAfter a continuous-time signal has been through the A/D converter, the quantized

output may differ from the input value. This deviation from the ideal output value is called the quantization error.

25. What are recursive and non-recursive systemIn Recursive systems, the output depends upon past, present, future value of inputs

as well as past output. In Non-Recursive systems, the output depends only upon past, present or future values of inputs.Example y(n)= x(n) + y(n-2) is recursive system and Y(n) = x(n) + x(n-1) is non recursive system.

26. Explain the frequency relationships between continuous time and discrete time signals.

Continuous time frequencies are given as Ω and F. while discrete time frequencies are given as ω and f. Conversion relationships are given as ω = Ω Ts and f=FTs.

27. What is the use of correlation in DSP. How it is related with linear convolution.

Correlations are nothing but establishing similarity between one set of data and another. Correlation is closely related to convolution, because the correlation is essentially convolution of two data sequences in which one of the sequences has been reversed.Applications are in 1) Images processing for (in which different images are compared)

2) In radar and sonar systems for range and position finding in which transmitted and reflected waveforms are compared. 3) Correlation is also used in detection and identifying signals in noise.

28. What is the relationship between difference equation and system function. System function can be obtained by taking Z transform of the difference equation.

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UNIT II, III Z TRANSFORM

1. What is Z transform and ROC. What is the usefulness of ROC. What are the applications of Z Transform.

For analysis of continuous time LTI system Laplace transform is used. And for analysis of discrete time LTI system Z transform is used. Z transform is mathematical tool used for conversion of time domain into frequency domain (z domain) and is a function of the complex valued variable Z. The z transform of a discrete time signal x(n) denoted by X(z) and given as

∞X(z) = ∑ x (n) z –n z-Transform.……(1) n=-∞

Z transform is an infinite power series because summation index varies from -∞ to ∞. But it is useful for values of z for which sum is finite. The values of z for which f (z) is finite and lie within the region called as “region of convergence (ROC).ADVANTAGES OF Z TRANSFORM : For calculation of DFT, for analysis and synthesis of digital filter, used for linear filtering, used for finding Linear convolution, cross-correlation and auto-correlations of sequences.ADVANTAGES OF ROC: ROC is going to decide whether system is stable or unstable, the type of sequences causal or anti-causal & decides finite or infinite duration sequences.

2. How poles and zeros & ROC decides the causality and stability of system. LSI system is stable if and only if the ROC the system function includes the unit

circle. i.e r < 1. Thus Poles inside unit circle gives stable system. Poles outside unit circle gives unstable system. Poles on unit circle give marginally stable system.

LSI system is causal if and only if the ROC the system function is exterior to the circle. i. e |z| > r.

3. Discuss the nature of the signal.

4. Discuss the ROC of the signal & pole-zero plot of the signal. Consider two cases case 1: Infinite signal & case 2: Finite signal.

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5. Discuss the nature of the signal.

6. ROC does not contains poles. Discuss the correctness of this statement.

7. Define pole and zero of the system. What poles and zeros are plotted with respect to unit circle in z plane. The frequency at which the magnitude of transfer function approaches infinity is

called pole and the frequency at which magnitude of transfer function becomes zero is called zero. Unit circle is the frequency axis in z plane.

8. What is the use of Unilateral Z transform. Unilateral Z transform is used to solve the difference equation.

9. Can a pole and zero lie on the same point.

10. What are Dirichlet conditions.

11. Explain JURY'S Stability AlgorithmJury's stability algorithm says

1. Form the first rows of the table by writing the coefficients of D(z). B0 B1 B2 ---------BN

BN BN-1 BN-2 --------- B0

2. Form third and fourth rows of the table by evaluating the determinant C J

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3. This process will continue until you obtain 2N-3 rows with last two having 3 elements. Y0,Y1,Y2

A digital filter with a system function H(z) is stable, if and only if it passes the following conditions. a. D(Z)|Z=1 > 0b. (-1)N D(Z)|Z=-1 >0c. |b0|>|bN|, |C0|>|CN-1|

Z Transform Properties

Sr No Property X(n) X(z)1 Linearity a1 x1(n) + a2 x2(n) a1 X1(z) + a2 X2(z)2 Time shifting x(n-k) X(z) z–k

3 Scaling in z domain an x(n) x(z/a)4 Time reversal x(-n) x(z-1)5 Convolution

Theoremx1(n) * x2(n) X1(z) X2(z)

Standard Z Transforms Sr No X(n) Property X(Z) ROC1 δ(n) 1 complete z plane2 δ(n-k) Time shifting z-k except z=03 δ(n+k) Time shifting zk except z=∞4 u(n) 1/1- z-1 |z| > 15 u(-n) Time reversal 1/1- z |z| < 16 -u(-n-1) Time reversal z/z- 1 |z| < 17 n u(n) Differentiatio

nz-1 / (1- z-1)2 |z| > 1

8 an u(n) Scaling 1/1- (az-1) |z| > |a|9 -an u(-n-1) 1/1- (az-1) |z| < |a|10 n an u(n) Differentiatio

na z-1 / (1- az-1)2 |z| > |a|

11 -n an u(-n-1) Differentiation

a z-1 / (1- az-1)2 |z| < |a|

12 cos(ω0n) u(n) 1- z-1cosω0

1- 2z-1cosω0+z-2|z| > 1

13 sin(ω0n) u(n) z-1sinω0

1- 2z-1cosω0+z-2|z| > 1

B0 BN-J

BN BJ

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UNIT IV: FT,DFT AND FFT

1. Why the frequency domain analysis is preferred over time domain analysis in DSP.

Time domain analysis provides some information like amplitude at sampling instant but does not convey frequency content & power, energy spectrum hence frequency domain analysis is used. Magnitude and phase plot can be obtained from its FT and system characteristic can be described well by using its frequency domain.

2. What is DTFT. Explain the nature of the spectrum of discrete time signal. The discrete time Fourier transform of the signal is denoted as X(ω). It is also called

as analysis equation. It is given as ∞

X(ω) = ∑ x (n) e –jωn

n=-∞Here ω is the frequency of discrete time signal and it takes all possible values between -∏ to ∏. Hence its Fourier transform is continuous. Case 1: If x(n) is infinite or finite non-periodic sequence then its spectrum X(ω) is continuous in nature. Case 2: If x(n) is finite periodic sequence then its spectrum X(ω) will be discrete. Inverse DTFT is also called synthesis equation. Here integration is used since X(ω) is the continuous function of ω. Integration limits are -∏ to ∏. And the period of integration is 2∏.

3. What is the existence criteria of DTFT. Why it is used. In the definition of DTFT, there is summation over infinite range of n. Hence for DTFT

to exist, the convergence of this summation is necessary. Hence existence criteria is ∞

∑ |x(n)| < ∞ n=-∞

IDTFT does not have convergence problem since the integration is over limited range.

4. What are the symmetry properties of FT.

Sr No Sequence

DTFT

1 X*(n) X*(- ω)

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2 X*(-n) X*(ω)3 XR(n) Xe(ω)=1/2 [ X(ω) + X*(-ω)]4 jXI(n) Xo(ω)=1/2 [ X(ω) - X*(-ω)]5 Xe(n) XR(ω)6 Xo(n) jXI(ω)

DTFT Properties:

Sr No Property Time domain Sequence

Frequency Domain Sequence

1 Periodicity x(n) X(ω+2∏k)= X(ω)

2 Linearity a1x1(n)+a2x2(n) a1X1(ω)+a2X2(ω)

3 Time Shifting x(n-k) e-jωk X(ω)

4 Time Reversal x(-n) X(-ω)

5 Convolution x1(n) * x2(n) X1(ω)+ X2(ω)

6 Frequency Shifting e-jωon x(n) X(ω- ω0)

7 Scaling x(pn) X(ω/p)

8 Differentiation -j n x(n) d/dω [X(ω)]

9 Parseval's Theorem

Energy of the signal is given byE= 1/2∏ ∫ |X(ω)|2 dω

DFT Properties:

Sr No

Property Time domain Sequence

Frequency Domain Sequence

1 Periodicity x(n) X(k+N)= X(k)

2 Linearity a1x1(n)+a2x2(n) a1X1(k)+a2X2(k)

3 Circular Time Shift X((n-k))N e-j2∏kl/N X(k)

4 Time Reversal X((-n))N X((-k))N

5 Circular Convolution x1(n) N x2(n)N-1∑ x1(n) x2((m-n))N

n=0

6 Circular frequency Shifting ej2∏kl/N X(n) X((k-l))N

7 Parseval's TheoremEnergy of the signal is given by N-1E= 1/N ∑ |X(k)|2 K=0

5. Why DFT's are used in frequency domain analysis in place of DTFT. FT is the continuous function of x(n) and the range of ω is from - ∏ to ∏ or 0 to 2∏.

while DFT is calculated only at discrete values of ω. Thus DFT is discrete in nature which is sampling version of FT and thus mostly used in analysis of discrete signals.

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For Discrete time signals x(n) , Fourier Transform is denoted as x(ω) & given by ∞

X(ω) = ∑ x (n) e –jωn

n=-∞DFT is denoted by x(k) and given by (ω= 2 ∏ k/N)

N-1X(k) = ∑ x (n) e –j2 ∏ kn / N

n=0

6. Circular convolution and Linear convolution are same or different. Multiplication of two sequences in time domain is called as Linear convolution while

Multiplication of two sequences in frequency domain is called as circular convolution. They are one and same but they differ in total number of samples in it.

7. What are overlap save and add method. Why these methods are used. When the input data sequence is long then it requires large time to get the output

sequence. Hence other techniques are used to filter long data sequences. Instead of finding the output of complete input sequence, it is broken into small length sequences. The output due to these small length sequences are computed fast. The outputs due to these small length sequences are fitted one after another to get the final output response.

8. What is FFT. In which applications it is preferred over DFT.Large number of the applications such as filtering, correlation analysis, spectrum

analysis require calculation of DFT. But direct computation of DFT require large number of computations and hence processor remain busy. Hence special algorithms are developed to compute DFT quickly called as Fast Fourier algorithms (FFT).

The radix-2 FFT algorithms are based on divide and conquer approach. In this method, the N-point DFT is successively decomposed into smaller DFT’s. Because of this decomposition, the number of computations are reduced.

9. If input signal x(n) contains 4 samples. How many samples will be present in its DFT. What will happen if it contains less than 4 samples.

10. What is the difference between DITFFT and DIFFFT.Sr No

DIT FFT DIF FFT

1 DITFFT algorithms are based upon decomposition of the input sequence into smaller and smaller sub sequences.

DIFFFT algorithms are based upon decomposition of the output sequence into smaller and smaller sub sequences.

2 In this input sequence x(n) is splitted into even and odd numbered samples

In this output sequence X(k) is considered to be splitted into even and odd numbered samples

3 Splitting operation is done on time domain sequence.

Splitting operation is done on frequency domain sequence.

4 In DIT FFT input sequence is in bit reversed order while the output sequence is in natural order.

In DIFFFT, input sequence is in natural order. And DFT should be read in bit reversed order.

11. What is use of Goertzel Algorithm. If DFT is to be calculated at selected points only then, Goertzel algorithms are used.

Goertzel algorithms are used to calculated DFT as linear filtering operations and required less number of calculations.

12. State the relationship between ZT and FT. There is a close relationship between Z transform and Fourier transform. If we

replace the complex variable z by e–jω, then Z transform is reduced to Fourier transform.

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13. What mathematical tools are used to convert the signals from time domain to frequency domain.

14. What are Dirichlet conditions.

15. Expansion in time domain is equivalent to compression in frequency domain. Discuss this statement with an example.

16. If two sequences are multiplied in time domain what will be effect on their DFT's.

17. Circular Convolution can be obtained from linear convolution but vice-versa is not possible. Discuss this statement with an example.

18. State any two applications of A) Linear convolution B)Circular Convolution C) DFT D) FFT

19. What is use of bit reversal technique. Where it is used.

Decimal

Memory Address x(n) in binary (Natural Order)

Memory Address in bit reversed order

New Address in decimal

0 0 0 0 0 0 0 01 0 0 1 1 0 0 42 0 1 0 0 1 0 23 0 1 1 1 1 0 64 1 0 0 0 0 1 15 1 0 1 1 0 1 56 1 1 0 0 1 1 37 1 1 1 1 1 1 7

Table shows first column of memory address in decimal and second column as binary. Third column indicates bit reverse values. As FFT is to be implemented on digital computer simple integer division by 2 method is used for implementing bit reversal algorithms.

20. Explain In Place computation and Memory requirement concept. a A= a + WN

r b

b WNr B= a - WN

r b

From values a and b new values A and B are computed. Once A and B are computed, there is no need to store a and b. Thus same memory locations can be used to store A and B where a and b were stored hence called as In place computation. The advantage of in place computation is that it reduces memory requirement. Thus for computation of one butterfly, four memory locations are required for storing two complex numbers A and B.

21. Can FFT Algorithms are applicable for the values of N which are not power of 2. Example N=12.Yes, In such cases sequence is padded with sufficient number of zeros such that the

value of N becomes the power of 2. Alternately (Another method)If N=12, It can be divided into 3 sequence of 4 samples each. These sequences will be as followsFirst Sequence: x(0), x(3), x(6), x(9)Second sequence: x(1), x(4), x(7), x(10)Third sequence: x(2), x(5), x(8), x(11)Now 4 point DFT's are calculated and then proceed further.

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UNIT V DIGITAL FILTER

1. What is Sinc function. Sinc pulse represents impulse response of ideal LPF while impulse train

represents ideal sampling function.

2. What is inversibility property. If the system is invertible then HH-1=1. This means if the two systems are cascaded,

output is same as input. Thus the condition for system to be invertible in terms of impulse response is h(n)*h-1(n) = δ(n).

3. Difference between analog and digital filter.Analog filters are used for filtering analog signals while digital filters are used for

digital signals. Analog filters are designed with various components like resistor, inductor and capacitor and digital Filters are designed with digital hardware like FF, counters shift registers, ALU and software’s like C or assembly language.

Digital filters are more accurate, less sensitive to environmental changes, most flexible, programmable and stable.

4. What are ideal filter characteristic.1. Ideal filters have a constant gain (usually taken as unity gain) passband

characteristic and zero gain in their stop band.2. Ideal filters have a linear phase characteristic within their passband.3. Ideal filters also have constant magnitude characteristic.

5. What are notch and Comb filters. What are its applications.A notch filter is a filter that contains one or more deep notches or ideally perfect

nulls in its frequency response characteristic. Notch filters are useful in many applications where specific frequency components must be eliminated. Example Instrumentation and recording systems required that the power-line frequency 60Hz and its harmonics be eliminated.

comb filters are similar to notch filters in which the nulls occur periodically across the frequency band similar with periodically spaced teeth. Frequency response characteristic of notch filter |H(ω)| is as shown

ωo ω1 ω

6. What are digital resonators. In which applications they are used. A digital resonator is a special two pole bandpass filter with a pair of complex

conjugate poles located near the unit circle. The name resonator refers to the fact that the filter has a larger magnitude response in the vicinity of the pole locations. Digital resonators are useful in many applications, including simple bandpass filtering and speech generations.

7. What is difference between FIR and IIR filterFIR system has finite duration unit sample response. i.e h(n)=0 for n<0 and n ≥ M

IIR system has infinite duration unit sample response. i. e h(n) = 0 for n<0

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FIR systems are non recursive. Thus output of FIR filter depends upon present and past inputs while IIR systems are recursive. Thus output of IIR filter depends upon present and past inputs as well as past outputs.

FIR filters are most stable, requires limited memory. In IIR filters stability can not be guaranteed and requires infinite memory.

8. In which applications FIR filters are designed.FIR filters can have an exactly linear phase response so that no phase distortion is

introduced in the signal by the filter. Hence FIR filters are generally used if no phase distortion is desired. Example: Data Transmission over a long distance and speech processing FIR filters are used.

9. In which applications IIR filters are designed.IIR filters are generally used if sharp cutoff and high throughput is required. Also

Analogue filters can be easily and readily transformed into equivalent IIR digital filter.

10. How the stable filters can be designed.All poles should be placed inside the unit circle on order for the filter to be stable.

However zeros can be placed anywhere in the z plane. 1. FIR filters are all zero filters hence they are always stable. 2. IIR filters are stable only when all poles of the filter are inside unit circle.

11. Difference between impulse invariance and BZT method.Impulse invariance: In this method IIR filters are designed having a unit sample response h(n) that is sampled version of the impulse response of the analog filter. Hence small value of T is selected to minimize the effect of aliasing. Frequency relationship is linear and all poles are mapped But the main disadvantage of this method is that it does not correspond to simple algebraic mapping of S plane to the Z plane. Thus the mapping from analog frequency to digital frequency is many to one.

Bilinear transformation Method: The bilinear transformation is a conformal mapping that transforms the j Ω axis into the unit circle in the z plane only once, thus avoiding aliasing of frequency components. But Frequency relationship is non-linear. Frequency warping or frequency compression is due to non-linearity.

Impulse invariance method is generally used for designing low frequencies filter like LPF. while for designing of LPF, HPF and almost all types of Band pass and band stop filters BZT method is used.

12. Plot Mapping between analog and digital filter frequencies in BZT method.

ω 2 tan -1 (ΩT/2)

ΩT

13. What is frequency warping. Why it is used in filter design.

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In BZT Frequency relationship is non-linear. Frequency warping or frequency compression is due to non-linearity. Frequency warping means amplitude response of digital filter is expanded at the lower frequencies and compressed at the higher frequencies in comparison of the analog filter. But the main disadvantage of frequency warping is that it does change the shape of the desired filter frequency response.

14. What are different approximation. how it is useful in filter design.No Practical filters can provide the ideal characteristic. Hence approximation of the

ideal characteristic are used. Such approximations are standard and used for filter design. Such three approximations are regularly used. Butterworth Filter Approximation, Chebyshev Filter Approximation and Elliptic Filter Approximation

Butterworth filters are defined by the property that the magnitude response is maximally flat in the passband.

15. State the mapping between Z Plane and S plane in Impulse Invariance method or Bilinear Transformation method. 1) Left side of s-plane is mapped inside the unit circle.2) Right side of s-plane is mapped outside the unit circle.3) jΩ axis is in s-plane is mapped on the unit circle.

Im[z] jΩ

Re(z) σ

Z-Plane S-Plane

16. What is all pass filter. What are its applications.An all pass filter is defined as a system that has a constant magnitude response for

all frequencies.|H(ω)| = 1 for 0 ≤ ω < ∏

The simplest example of an all pass filter is a pure delay system with system function H(z) = Z-k. This is a low pass filter that has a linear phase characteristic.

All Pass filters find application as phase equalizers. When placed in cascade with a system that has an undesired phase response, a phase equalizers is designed to compensate for the poor phase characteristic of the system and therefore to produce an overall linear phase response. 17. FIR filter are always stable. Explain.In FIR Impulse response of the system is given as

H(n) = bn for 0 ≤ n ≤ M-1 = 0 otherwise.

i.e Y(n) = b0 x(n) + b1 x(n-1) + …….. + bM-1 x(n-M+1) Thus y(n) is bounded if input x(n) is bounded. This means FIR system produces bounded output for every bounded input. Hence FIR systems are always stable.

18. What are the various method used for FIR & IIR filter designThe various methods used for IIR Filer design are as follows1. Approximation of derivatives2. Impulse Invariance3. Bilinear Transformation The various method used for FIR Filer design are as follows

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3

1

2

1. Windowing Method2. DFT method3. Frequency sampling Method. (IFT Method)

19. What are Gibbs phenomenonImpulse response of an ideal LPF is as shown in Fig.

In Fourier series method, limits of summation index is -∞ to ∞. But filter must have finite terms. Hence limit of summation index change to -Q to Q where Q is some finite integer. But this type of truncation may result in poor convergence of the series. Abrupt truncation of infinite series is equivalent to multiplying infinite series with rectangular sequence. i.e at the point of discontinuity some oscillation may be observed in resultant series.

Consider the example of LPF having desired frequency response Hd (ω) as shown in figure. The oscillations or ringing takes place near band-edge of the filter. This oscillation or ringing is generated because of side lobes in the frequency response W(ω) of the window function. This oscillatory behavior is called "Gibbs Phenomenon".

20. Ideal filter are not physically realizable. Why.LSI system is causal if its unit sample response satisfies following condition.

h(n) = 0 for n<0In above figure h(n) extends -∞ to ∞. Hence h(n) ≠0 for n<0. This means causality condition is not satisfied by the ideal low pass filter. Hence ideal filters are anti-causal and thus are not physically realizable.

21. FIR Filters always provides linear phase response. Explain. The phase or angle of H(ω) is given as

-ω M-1 for |H (ω)| > 0 2

Angle H(ω) = -ω M-1 for |H (ω)| < 0 2

In above equations M is constant. Hence Phase of H(ω) is linear function of ω. That is phase is linearly proportional to frequency. When |H(ω)} changes sign, phase changes by ∏. Thus FIR filters are linear phase filters. This is important feature of FIR Filters.

22. For Speech processing or data transmission which type of filter are preferred.

FIR filter always provides linear phase response. This specifies that the signals in the pass band will suffer no dispersion Hence when the user wants no phase distortion, then FIR filters are preferable over IIR. Phase distortion always degrade the system performance. In various applications like speech processing, data transmission over long distance FIR filters are more preferable due to this characteristic. Another reason is that quantization noise can be made negligible in FIR filters.

23. How FIR filters can be classified.

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+∏

FIR filters can be classified into two types. Symmetric and Anti-symmetric FIR filters1 Unit sample response of FIR filters is symmetric if it satisfies following condition

h(n)= h(M-1-n) n=0,1,2…………….M-12. Unit sample response of FIR filters is Anti-symmetric if it satisfies following

conditionh(n)= -h(M-1-n) n=0,1,2…………….M-1

24. Why FIR needs higher orders for similar magnitude response compared to IIR filters.

Impulse response of ideal low pass filter is as shown in fig. In order to have finite terms we will multiply this infinite series with rectangular window which will generate desired frequency response. But some oscillation or ringing effect will be observed at the point of truncation. This effect is known as Gibbs Phenomenon.

As M increases this side lobes becomes narrow and oscillatory behavior decreases. As an example, the impulse response for a LPF is truncated with M=9,25 and an infinite number of samples is as shown.

25. What are Windows techniques? How they are selected. Impulse response of ideal filter is infinite but in FIR filter, h(n) is finite. Hence in order

to truncate infinite impulse response to finite range we will multiply it to window and thus practically implemented. There are various types of windows like rectangular, triangular, hamming, Hanning window etc. The windows are selected depending upon the transition width of main lobe and amplitudes of sidelobes. The windows are selected such that Gibb's phenomenon is reduced.

The particular window is selected depending upon minimum stop band attenuation.

26. What are different window functions used for design of FIR filters.Different types of windows functions are available which reduce ringing effect. These

are Triangular window, Blackman, Hamming window, Hanning Window and Kaiser window. a. FIR filters designed using hamming window has reduced sidelobes compared to

rectangular window. b. Blackman window has very small sidelobes but increased width of main lobe. In

Kaiser window has reduced side lobes and transition band is narrow and hence mostly used.

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27. What are the constraints to be imposed while designing filters from it pole zero plot.Filters can be designed from its pole zero plot. Following two constraints should be imposed while designing the filters.1. All poles should be placed inside the unit circle on order for the filter to be stable. However zeros can be placed anywhere in the z plane. FIR filters are all zero filters hence they are always stable. IIR filters are stable only when all poles of the filter are inside unit circle.2. All complex poles and zeros occur in complex conjugate pairs in order for the filter coefficients to be real.In the design of low pass filters, the poles should be placed near the unit circle at points corresponding to low frequencies ( near ω=0)and zeros should be placed near or on unit circle at points corresponding to high frequencies (near ω=∏). The opposite is true for high pass filters.

28. Which window is better. Short duration window or long duration window.Long Duration window. Because the length of window must be infinite in ideal case.

29. What are frequency transformation techniques. Why they are used. Frequency transformation techniques are used to generate High pass filter, Bandpass and bandstop filter from the lowpass filter system function. Sr No

Type of transformation Transformation ( Replace s by)

1 High Passωhp

sωhp = Password edge frequency of HPF

2 Band Pass

(s2 + ωl ωh )s (ωh - ωl )

ωh - higher band edge frequencyωl - Lower band edge frequency

3 Band Stop

s (ωh - ωl)s2+ ωh ωl

ωh - higher band edge frequencyωl - Lower band edge frequency

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UNIT VI: DSP PROCESSOR

1. What are the requirements of DSP processor. How It differs from general Processor.The most fundamental mathematical operation in DSP is sum of products also called as dot of products.

Y(n)= h(0)*x(n) + h(1)*x(n-1) +………+ h(N-1)*x(n-N) This operation is mostly used in digital filter designing, DFT, FFT and many other DSP applications. A DSP is optimized to perform repetitive mathematical operations such as the dot product. There are four basic requirements of DSP processor to optimize the performance They are 1) Fast arithmetic 2) Fast Execution - Dual operand fetch 3) Fast data exchange4) Circular buffering

Sr No

Requirements Features of DSP processor

1 Fast Arithmetic Faster MACs means higher bandwidth.Able to support general purpose math functions, should have ALU and a programmable shifter function for bit manipulation. Powerful interrupt structure and timers

2 Fast Execution Parallel Execution is required in place of sequential. Instructions are executed in single cycle of clock called as True instruction cycle as oppose to multiple clock cycle.Multiple operands are fetched simultaneously. Multi-processing Ability and queue, pipelining facilityAddress generation by DAG's and program sequencer.

3 Fast data Exchange Multiple registers, Separate program and data memory and Multiple operands fetch capacity

4 Circular shift operations Circular Buffers

2. What are different microprocessor architectures. Which is mostly used in DSP processor. There are mainly three types of microprocessor architectures present. 1. Von-Neumann architecture2. Harvard architecture3. Analog devices Modified Harvard architecture.

Harvard Architecture is common to many DSP processors. The processor can simultaneously access two memory blanks using two independent sets of buses allowing

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operands to be loaded while fetching instructions. Von-Neumann memory architecture is common among microcontrollers Since there is only one data bus, operands can not be loaded while instructions are fetched.

3. Explain core architecture of ADSP-21xx processor.ADSP-21xx family DSP's are used in high speed numeric processing applications. ADSP-21xx architecture consists of

Five Internal BusesProgram Memory Address(PMA)Data memory address (DMA)Program memory data(PMD)Data memory data (DMD)Result (R)

Three Computational UnitsArithmetic logic unit (ALU)Multiply-accumulate (MAC)Shifter

Two Data Address generators (DAG) Program sequencer On chip peripheral Options

RAM or ROMData Memory RAMSerial PortTimerHost Interface PortDMA Port

FEATURES OF ADSP-21xx PROCESSOR1. 16 bit fixed DSP microprocessor2. Enhanced Harvard architecture for three bus performance.3. Separate on chip buses for program and data memory.

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4. 25 MIPS, 40 ns maximum instruction set 25Mhz frequency.5. Single cycle instruction execution i.e True instruction cycle.

4. What are ADSP-21xx Development toolsVarious development tools such as assembler , linker, debugger, and simulator are available for ADSP-21xx family.

The system builder is the software development tool for describing the configuration of the target system's memory and I/O. The ranges for program memory(PM) and data memory(DM) are described.

The assembler translated source code into object code modules. The source code is written in assembly language file (.DSP) Assembler reads .DSP file and generates fouroutput filed with the same root name. Object file(.OBJ), Code File(.CDE), Initialization File (.INT), List File(.LST) etc.

The linker is a program used to join together object files into one large object file. The linker produces a link file which contains the binary codes for all the combined modules.

A debugger is a program which allows user to load object code program into system memory, execute the program and debug it.

Difference between DSP and General Purpose Processor

Sr

Parameter General Purpose Processor

DSP Processor

1 Instruction Cycle Multiple clock cycles required for execution of one instruction

Single cycle of the clock is needed.

2 Instruction execution

Sequential Execution Parallel execution - Pipelining involved

3 Operand fetch from the memory

Sequential Multiple operand fetch capability

4 Memories No separate memory Separate program and data memory

5 Instruction set Mostly Contains Data movement instructions

Contains complex addition, multiplication & shifting instructions

6 Address generation

PC is used DAG and Program Sequencer

7 On chip address and data buses

Single pair of buses PMA,DMA, PMD and DMD

8 Computational Units

ALU ALU, MAC and Shifter

5. What are the different functions used in MATLAB related with DSP.

Sr No Function Application1 conv(hn,xn) Linear Convolution of two sequences.2 xcorr(x1n,x2n) Cross Correlation of two sequences.3 xcorr(x) Auto correlation of sequence3 fft(xn) DFT of x(n) using FFT algorithm4 ifft(xn) IDFT of x(n) using FFT algorithm5 Overlpsav (x,h,N) Implement Overlap save method to perform block

convolution. 5 zplane (b,a) Plot Pole zero plot.6 freqz (b,a) Plot Magnitude phase plot.7 freqs (b,a) Compute the frequency response of an analog filter.8 bilinear(z,p,k,fs) Bilinear transformation9 boxvar (M) Rectangular Window

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10 hanning (M) Hanning Window11 hamming (M) Hamming Window12 kaiser(M) Kaiser Window

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