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Correct OPTION is (b) Cascade form

To explain: The design is most easily CARRIED out with the system function for the IIR filter EXPRESSED in the cascade form as

H(z)=G.A(z).

Correct ANSWER is (C) Toeplitz MATRIX

To elaborate: We observe that the matrix is not only symmetric but it also has the special property that all the elements ALONG any diagonal are equal. Such a matrix is CALLED a Toeplitz matrix and lends itself to efficient inversion by means of an algorithm.

The correct OPTION is (b) M^2

The best I can explain: The Levinson-Durbin ALGORITHM is the algorithm which is used for the efficient inversion of TOEPLITZ MATRIX which requires a number of computations PROPORTIONAL to M^2 instead of the usual M^3.

Correct answer is (a) True

For explanation I would say: The optimum, in the least SQUARE sense, FIR filter that satisfies the LINEAR equations in \(\sum_{k=0}^M b_k r_{hh} (k-l)=r_{dh} (l)\), l=0,1,…M is called the wiener filter.

The correct ANSWER is (d) all of the mentioned

Best explanation: When ε is minimized with respect to the filter coefficients, we obtain the set of LINEAR equations

\(\sum_{k=0}^M b_k r_{hh}(k-l)=r_{dh} (l)\), l=0,1,…M

and we know that RDH(l) depends on the desired OUTPUT d(N).

Right CHOICE is (B) Least squares error criterion

Easy EXPLANATION: We can use the least squares error criterion to optimize the M+1 coefficients of the FIR filter.

Correct ANSWER is (a) True

For explanation: When ε is minimized with respect to the filter coefficients, we obtain the set of linear EQUATIONS which are dependent on the auto CORRELATION sequence of the signal h(n).

Correct ANSWER is (d) Least SQUARES wiener FILTER

Best explanation: Since from the BLOCK diagram, the COEFFICIENTS of the FIR filter coefficients are optimized by the least squares error criterion, it belongs to the least squares FIR inverse filter or wiener filter.

The correct choice is (C) M+1

To explain I WOULD SAY: In the process of TRUNCATING, we INCUR a total squared approximation error where M+1 is the length of the truncated filter.

Correct choice is (C) H(z).HI(z)=1

Easiest EXPLANATION: The inverse to a linear time INVARIANT system with impulse response h(N) and system function H(z) is defined as the system whose impulse response is hI(n) and system function HI(z), satisfy the following condition

H(z).HI(z)=1.

The correct ANSWER is (B) False

Easy explanation: In most of the PRACTICAL applications, it is desirable to RESTRICT the inverse filter to be an FIR filter.

The correct option is (a) True

Easy EXPLANATION: We find the coefficients {bk} by pade APPROXIMATION and find the coefficients {ak} by least squares method. Thus the foregoing approach for determining the POLES and zeros of H(Z) is sometimes CALLED as Prony’s method.

The correct ANSWER is (a) {bk}

To elaborate: The parameters {bk} are selected to DETERMINE the zeros of the FILTER that can be obtained where h(n)=HD(n).

The correct choice is (c) N > M

The best I can explain: Nevertheless, we can use the desired RESPONSE HD(n) for n < M to construct an estimate of hd(n).

Correct OPTION is (a) True

Explanation: We know that we can perform pole-zero approximation for HD(z) by using the least squares METHOD.

Correct choice is (d) Correlation

For EXPLANATION I would say: In a PRACTICAL DESIGN problem, the desired impulse response HD(n) is specified for a FINITE set of points, say 0 < n

The correct answer is (B) False

To elaborate: By differentiating the square of the ERROR sequence with respect to the parameters {ak}, it is EASILY ESTABLISHED that we obtain the set of linear equations.

The CORRECT answer is (b) {ak}

EASIEST explanation: The PARAMETERS {ak} are selected to MINIMIZE the sum of squares of the error sequence.

The CORRECT answer is (C) 1

To ELABORATE: The condition that yd(0)=y(0)=1 is SATISFIED by selecting b0=hd(0).

Correct ANSWER is (a) δ(N)

The best I can explain: We excite the cascade CONFIGURATION by the unit sample sequence δ(n). Thus the input to the INVERSE system 1/H(z) is hd(n) and the OUTPUT is y(n). Ideally, yd(n)= δ(n).

Correct option is (B) All pole

The best I can explain: Let us assume that hd(n) is SPECIFIED for n > 0, and the digital filter is an all-pole filter.

The CORRECT answer is (c) Type-II chebyshev FILTER

The explanation is: The DIAGRAM that is GIVEN in the question is the impulse response of type-II chebyshev filter.

Right answer is (d) Hd(Z), 1/H(z)

Best EXPLANATION: In this method, we consider the CASCADE connection of the DESIRED filter Hd(z) with the RECIPROCAL, all zero filter 1/H(z).

Right answer is (d) 4,5

For explanation: When M is increased from three to four, we obtain a PERFECT match with the desired BUTTERWORTH FILTER not only for N=4 but for N=5, and in fact, for LARGER values of N.

The CORRECT CHOICE is (a) 3

The explanation: We OBSERVE that when the NUMBER of zeros in minimum, that is when M=3, the resulting frequency response is a relatively poor approximation to the desired response.

The CORRECT option is (d) All of the mentioned

The explanation is: We obtain a PERFECT match between h(N) and the desired response hd(n) for the FIRST L values of the impulse response and we also know that L=M+N+1.

The correct choice is (b) Both POLES and zeros

Explanation: The MAJOR limitation of Pade approximation METHOD, namely, the resulting filter must CONTAIN a large number of poles and zeros.

Right option is (c) L-1

Explanation: If we SELECT the upper LIMIT as U=L-1, it is possible to match H(n) PERFECTLY to the desired response hd(n) for 0 < n < M+N.

Right option is (a) True

Easy explanation: In general, H(N) is a non-linear function of the filter parameters and hence the minimization of ε involves the solution of a SET of non-linear equations.

Correct ANSWER is (a) M+N+1

To EXPLAIN I would say: The filter has L=M+N+1 PARAMETERS, NAMELY, the coefficients {ak} and {BK}, which can be selected to minimize some error criterion.

Correct option is (d) All of the mentioned

For EXPLANATION I would say: There are several methods for designing DIGITAL filters directly. The three techniques are Pade approximation and least square method, the specifications are given in the time domain and the design is CARRIED in time domain. The other ONE is least squares technique in which the design is carried out in frequency domain.

Right CHOICE is (B) Impulse invariance

Explanation: Except for the impulse invariance method, the design techniques for IIR filters INVOLVE the conversion of an analog filter into a digital filter by some MAPPING from the s-plane to the z-plane.

Right ANSWER is (a) True

Best explanation: It is better to PERFORM the mapping from an analog low pass FILTER into a DIGITAL low pass filter by either of these mappings and then perform the frequency transformation in the digital domain because by this kind of frequency transformation, problem of aliasing is avoided.

Right choice is (c) Bilinear TRANSFORMATION

Best explanation: In the CASE of bilinear transformation, where aliasing is not a problem, it does not matter WHETHER the FREQUENCY transformation is performed in the analog domain or in frequency domain.

Right answer is (b) False

Easy explanation: Since there is a problem of aliasing in designing high pass and many BAND pass filters USING IMPULSE invariance and mapping of derivatives, we cannot employ the analog FREQUENCY transformation followed by conversion of the RESULT into digital domain by use of these two mappings.

The correct option is (a) True

The best explanation: We know that the impulse invariance METHOD and mapping of derivatives are inappropriate to use in the designing of HIGH pass and band pass filters DUE to aliasing PROBLEM.

Right option is (d) All-pass

The explanation: We know that the UNIT CIRCLE must be MAPPED inside the unit circle.

Thus it implies that for r=1, e^-jω = g(e^-jω)=|g(ω)|.e^j arg [ g(ω) ]

It is clear that we must have |g(ω)|=1 for all ω. That is, the MAPPING is all-pass.

Right ANSWER is (a) < 1

For explanation I WOULD SAY: The value of |ak| < 1 to ENSURE that a stable filter is transformed into another stable filter to SATISFY the condition to satisfy the condition 1.

Right option is (C) Impulse INVARIANCE & Mapping of derivatives

To ELABORATE: We know that the impulse invariance method and mapping of derivatives are INAPPROPRIATE to use in the designing of high PASS and band pass filters.

Right answer is (b) False

For explanation I WOULD say: For the map z^-1 → g(z^-1) to be a valid digital FREQUENCY transformation, then the unit circle ALSO must be mapped INSIDE the unit circle.

Right option is (C) INSIDE the unit circle

Easiest explanation: The map z^-1 → g(z^-1) MUST map inside the unit circle in the z-plane into itself to APPLY digital frequency TRANSFORMATION.