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The correct CHOICE is (a) True

Explanation: The parallel FORM realization is also KNOWN as normal form representation, because the matrix F is diagonal, and HENCE the state variables are uncoupled.

The CORRECT answer is (b) (F-λI)U=0

Explanation: A number λ is an Eigen VALUE of F and a nonzero vector U is the ASSOCIATED Eigen vector if

FU=λU

Thus, we obtain (F-λI)U=0.

Right answer is (a) True

Best explanation: We KNOW that (F-λI)U=0

The above equation has a NONZERO SOLUTION U if the matrix F-λI is singular, which is the case if the determinant of (F-λI) is ZERO. That is, |F-λI|=0.

This determinant yields the characteristic polynomial of the matrix F.

Correct option is (c) TRANSPOSED system

To explain: The TRANSPOSE of the MATRIX F is obtained by interchanging its rows and columns, and it is denoted by F^T. The system thus obtained is KNOWN as Transposed system.

Correct choice is (d) Diagonal

Easy explanation: A closed form solution of the state space equations is EASILY OBTAINED when the SYSTEM matrix F is diagonal. HENCE, by finding a matrix P so that F^1=PFP^-1 is diagonal, the solution of the state equations is simplified CONSIDERABLY.

The correct OPTION is (a) True

The BEST explanation: If H(n) is the impulse response of the single input-single output system, and h1(n) is the impulse response of the TRANSPOSED system, then we know that h(n)=h^1>(n). Thus, a single input-single output system and its TRANSPOSE have identical impulse responses and hence the same input-output relationship.

The CORRECT option is (b) False

For explanation: According to the definition of STATE of a system, the system consists of two components called MEMORY component and memory LESS component.

Right option is (a) True

Best EXPLANATION: The state variables provide information about all the internal SIGNALS in the system. As a RESULT, the state-space description PROVIDES a more DETAILED description of the system than the input-output description.

The correct CHOICE is (c) STATE variables

For EXPLANATION I would say: Although the state space or the internal description of the SYSTEM still involves a relationship between the INPUT and output signals, it also involves an additional set of variables, called State variables.

Correct choice is (b) TRANSPOSED form

For explanation: According to the transposition or flow-graph reversal theorem, if we reverse the directions of all branch transmittances and INTERCHANGE the input and OUTPUT in the flow graph, then the system REMAINS unchanged. The resulting structure is known as transposed structure or transposed form.

Right CHOICE is (a) True

To elaborate: The INPUT to a system originates at a source node and the OUTPUT signal is extracted at a SINK node.

Correct ANSWER is (d) Summing node

To EXPLAIN: Summing node is the node which is USED in the SIGNAL flow graph which replaces the adder in the structure of a filter.

The correct option is (a) True

Easiest explanation: A signal flow graph PROVIDES an alternative, but an EQUIVALENT graphical representation to a block diagram structure that we have been using to illustrate various system realization. The BASIC elements of a flow graph are branches and NODES.

Right option is (d) MAX [M,N]

For EXPLANATION I would say: From the direct form-II realization of the IIR filter, if M and N are the orders of NUMERATOR and denominator of rational system function RESPECTIVELY, then Max[M,N] memory locations are required.

The CORRECT option is (a) M+N+1

Best EXPLANATION: From the DIRECT form-I realization of the IIR FILTER, if M and N are the orders of NUMERATOR and denominator of rational system function respectively, then M+N+1 memory locations are required.

Correct choice is (B) False

The BEST I can explain: In DIRECT form-I realization, all-zero SYSTEM is PLACED before the all-pole system.

The CORRECT choice is (b) M+N

The best I can explain: From the direct form-I realization of the IIR FILTER, if M and N are the orders of NUMERATOR and denominator of rational system function respectively, then M+N additions are required.

The correct option is (c) (1/4,1/2,1/3)

The explanation is: Given the system FUNCTION of the FIR filter is

H(z)= 1+(13/24)z^-1+(5/8)z^-2+(1/3)z^-3

Thus the lattice coefficients corresponding to the given filter is (1/4,1/2,1/3).

Right option is (c) M+N+1

Explanation: From the direct form-I realization of the IIR filter, if M and N are the ORDERS of numerator and DENOMINATOR of rational system function RESPECTIVELY, then M+N+1 MULTIPLICATIONS are required.

The correct option is (d) (1,13/24,5/8,1/3)

To elaborate: We get the output from the third STAGE lattice filter as

A3(z)=1+(13/24)z^-1+(5/8)z^-2+(1/3)z^-3.

Thus the FIR filter COEFFICIENTS for the direct form structure are (1,13/24,5/8,1/3).

The correct answer is (a) True

For EXPLANATION: The equation of the output from the second stage lattice filter is given by

f2(n)= X(n)+K1(1+K2)x(n-1)+K2x(n-2)

In the above equation, the constants K1 and K2 are called as REFLECTION coefficients.

The correct answer is (C) K1(1+K2)

For explanation: The equation for the OUTPUT of an FIR filter represented in the direct FORM structure is GIVEN as

y(n)=x(n)+ α2(1)x(n-1)+ α2(2)x(n-2)

The output from the double stage LATTICE structure is given by the equation,

f2(n)= x(n)+K2(1+K2)x(n-1)+K2x(n-2)

By comparing the coefficients of both the equations, we get

α2(1)= K1(1+K2).

Right option is (d) X(n)+K1(1+K2)x(n-1)+K2x(n-2)

For explanation: When two SINGLE stage LATTICE filters are cascaded, then the output from the first filter is given by the equation

f1(n)= x(n)+K1x(n-1)

g1(n)=K1x(n)+x(n-1)

The output from the SECOND filter is obtained as

f2(n)=f1(n)+K2g1(n-1)

=x(n)+K2[K1x(n-1)+x(n-2)]+ K1x(n-1)

= x(n)+K1(1+K2)x(n-1)+K2x(n-2).

Correct choice is (b) x(n)+Kx(n-1)

EXPLANATION: The single stage lattice FILTER is as shown below.

Here both the inputs are excited and OUTPUT is SELECTED from the top branch.

Thus the output of the single stage lattice filter is GIVEN by y(n)= x(n)+Kx(n-1).

Right choice is (a) True

Easiest explanation: The FIR STRUCTURE SHOWN in the above figure is INTIMATELY related with the topic of linear prediction. Thus the top filter structure shown in the above figure is called a prediction error filter.

Right option is (B) \(1+\sum_{K=1}^m α_m (k)Z^{-k}\)

Explanation: Consider a sequence of FIR FILER with system function HM(z)=Am(z), m=0,1,2…M-1

where, by definition, Am(z) is the polynomial

Am(z)=\(1+\sum_{k=1}^m α_m (k)z^{-k}\), m≥1 and A0(z)=1.

The correct choice is (d) FIR FILTER

To explain: The system function of the FIR filter according to the frequency sampling REALIZATION is given by the equation

H(Z)=\(\FRAC{1}{M}(1-z^{-M} e^{j2πα})\sum_{k=0}^{M-1}\frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)

The above system function can be represented in the cascade form as shown in the above block diagram.

Correct ANSWER is (c) HM(0)=1 and hm(k)=αm(k), k=1,2…m

The BEST explanation: We know that Hm(z)=Am(z) and Am(z) is a polynomial whose equation is given as Am(z)=\(1+\sum_{k=1}^m α_m (k)z^{-k}\), m≤1 and A0(z)=1

A0(z)=1 => hm(0)=1 and Am(z)=\(\sum_{k=1}^m α_m (k)z^{-k}\)(m≤1)=> hm(k)=αm(k) for k=1,2…m.

Right ANSWER is (d) All of the mentioned

The explanation is: Lattice filters are used EXTENSIVELY in digital signal processing and in the IMPLEMENTATION of adaptive filters.

Right option is (B) e^j2π(k+α)/M

For explanation I would SAY: The system function of the SECOND filter in the cascade of an FIR realization by frequency sampling method is given by

H2(Z)=\(\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)

We obtain the poles of the above system function by equating the denominator of the above EQUATION to zero.

=>\(1-e^{\frac{j2π(k+α)}{M}} z^{-1}\)=0

=>z=pk=\(e^{\frac{j2π(k+α)}{M}}\), k=0,1….M-1

Correct option is (C) \(\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)

Easiest explanation: The system function H(z) which is characterized by the set of FREQUENCY samples is obtained as

H(z)=\(\frac{1}{M}(1-z^{-M} e^{j2πα})\sum_{k=0}^{M-1}\frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}}z^{-1}}\)

We view this FIR realization as a cascade of two filters, H(z)=H1(z).H2(z)

Here H1(z) REPRESENTS the all-zero FILTER or comb filter, and the system function of the other filter is given by the equation

H2(z)=\(\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)

The correct choice is (b) On UNIT circle

For explanation: The system function of the comb filter is given by the equation

H1(z)=\(\FRAC{1}{M}(1-z^{-M} E^{j2πα})\)

Its zeros are located at equally spaced POINTS on the unit circle at

zk=e^j2π(k+α)/M k=0,1,2….M-1

Right option is (a) Two

For explanation: In frequency sampling realization, the system function H(z) is characterized by the set of frequency samples {H(k+ α)} INSTEAD of {h(N)}. We VIEW this FIR filter realization as a cascade of two filters. One is an all-zero or a comb filter and the other consists of parallel bank of single POLE filters with resonant frequencies.

Right ANSWER is (d) 50%

To explain I would say: We have to do 3 multiplications for every second order equation. So, we have to do 6 multiplications if we combine two second order EQUATIONS and we have to perform 3 multiplications by directly CALCULATING the FOURTH order equation. Thus the number of multiplications are reduced by a FACTOR of 50%.

Correct option is (C) \(\FRAC{2\pi}{M}\)(k+α)

Easiest EXPLANATION: To derive the frequency sampling structure, we specify the desired frequency RESPONSE at a SET of equally spaced frequencies, namely ωk=\(\frac{2\pi}{M}\)(k+α), k=0,1…(M-1)/2 for M odd

k=0,1….(M/2)-1 for M even

α=0 or 0.5.

The correct answer is (d) \(\frac{1}{M}(1-z^{-M} e^{j2πα})\)

Easy explanation: The system function H(z) which is characterized by the set of frequency samples is OBTAINED as

H(z)=\(\frac{1}{M}(1-z^{-M} e^{j2πα})\sum_{k=0}^{M-1}\frac{H(k+α)}{1-e^{j2π(k+α)/M} z^{-1}}\)

We view this FIR realization as a CASCADE of two filters, H(z)=H1(z).H2(z)

Here H1(z) REPRESENTS the all-zero filter or comb filter whose system function is given by the equation

H1(z)=\(\frac{1}{M}(1-z^{-M} e^{j2πα})\).

The correct answer is (b) M odd

The BEST explanation: When the FIR system has linear phase, the UNIT sample response of the system satisfies either the symmetry or asymmetry condition, h(n)=±h(M-1-n)

For such a system the number of multiplications is REDUCED from M to M/2 for M even and to (M-1)/2 for M odd. THUS for the structure GIVEN in the question, M is odd.

Right CHOICE is (a) True

Easiest EXPLANATION: The structure of the direct form REALIZATION, resembles a TAPPED DELAY line or a transversal system.

The correct option is (C) M-1

The explanation: The direct FORM realization follows immediately from the non-recursive difference equation given by y(n)=\(\sum_{k=0}^{M-1}b_k x(n-k)\).

We observe that this structure requires M-1 memory locations for STORING the M-1 previous INPUTS.

Right answer is (C) \(\sum_{k=0}^{M-1}b_k z^{-k}\)

Explanation: We KNOW that the DIFFERENCE equation of an FIR system is given by y(n)=\(\sum_{k=0}^{M-1}b_k x(n-k)\).

=>h(n)=BK=>\(\sum_{k=0}^{M-1}b_k z^{-k}\).

Right ANSWER is (a) True

For explanation: The DIFFERENCE equation y(n)=\(\sum_{k=0}^{M-1}b_k X(n-k)\) describes the FIR system.

Correct answer is (b) False

The best I can explain: APART from the three factors given in the question, other factors such as, whether the STRUCTURE or the realization lends itself to parallel processing or whether the COMPUTATIONS can be PIPELINED are also the factors which influence our choice of the realization of the system.

Right answer is (d) All of the mentioned

Easiest explanation: All the three of the CONSIDERATIONS GIVEN above are CALLED as finite word LENGTH effects.

Correct CHOICE is (a) True

For explanation I would say: The parameters of the system must necessarily be represented with finite precision. The computations that are performed in the process of computing an output from the system must be rounded off or truncated to fit within the limited precision constraints of the computer or HARDWARE used in the implementation. Thus, Finite word length effects refer to the quantization effects that are inherent in any DIGITAL implementation of the system, either in hardware or software.