1. If the frequency response of an FIR system is given as H(z)=1-z-1-6z-2, then the system is ___________
a) Minimum phase
b) Maximum phase
c) Mixed phase
d) None of the mentioned
Explanation: Given H(z)=1-z-1-6z-2
By factoring the system function we find the zeros for the system.
The zeros of the given system are at z=-2,3
So, the system is maximum phase.
2. If the frequency response of an FIR system is given as H(z)=1-5/2z-1-3/2z-2, then the system is ___________
a) Minimum phase
b) Maximum phase
c) Mixed phase
d) None of the mentioned
Explanation: Given H(z)= 1-5/2z-1-3/2z-2
By factoring the system function we find the zeros for the system.
The zeros of the given system are at z=-1/2, 3
So, the system is mixed phase.
3. An IIR system with system function H(z)=\(\frac{B(z)}{A(z)}\) is called a minimum phase if ___________
a) All poles and zeros are inside the unit circle
b) All zeros are outside the unit circle
c) All poles are outside the unit circle
d) All poles and zeros are outside the unit circle
Explanation: For an IIR filter whose system function is defined as H(z)=\(\frac{B(z)}{A(z)}\) to be said a minimum phase,
then both the poles and zeros of the system should fall inside the unit circle.
4. An IIR system with system function H(z)=\(\frac{B(z)}{A(z)}\) is called a mixed phase if ___________
a) All poles and zeros are inside the unit circle
b) All zeros are outside the unit circle
c) All poles are outside the unit circle
d) Some, but not all of the zeros are outside the unit circle
Explanation: For an IIR filter whose system function is defined as H(z)=\(\frac{B(z)}{A(z)}\) to be said a mixed phase and if the system is stable and causal, then the poles are inside the unit circle and some, but not all of the zeros are outside the unit circle.
5. A causal system produces the output sequence y(n)={1,0.7} when excited by the input sequence x(n)={1,-0.7,0.1}, then what is the impulse response of the system function?
a) [3(0.5)n+4(0.2)n]u(n)
b) [4(0.5)n-3(0.2)n]u(n)
c) [4(0.5)n+3(0.2)n]u(n)
d) None of the mentioned
Explanation: The system function is easily determined by taking the z-transforms of x(n) and y(n). Thus we have
H(z)=\(\frac{Y(z)}{X(z)} = \frac{1+0.7z^{-1}}{1-0.7z^{-1}+0.1z^{-2}} = \frac{1+0.7z^{-1}}{(1-0.2z^{-1})(1-0.5z^{-1})}\)
Upon applying partial fractions and applying the inverse z-transform, we get
[4(0.5)n-3(0.2)n]u(n).
6. If X(k) discrete Fourier transform of x(n), then the inverse discrete Fourier transform of X(k) is?
a) \(\frac{1}{N} \sum_{k=0}^{N-1}X(k)e^{-j2πkn/N}\)
b) \(\sum_{k=0}^{N-1}X(k)e^{-j2πkn/N}\)
c) \(\sum_{k=0}^{N-1}X(k)e^{j2πkn/N}\)
d) \(\frac{1}{N} \sum_{k=0}^{N-1}X(k)e^{j2πkn/N}\)
Explanation: If X(k) discrete Fourier transform of x(n), then the inverse discrete Fourier transform of X(k) is given as
x(n)=\(\frac{1}{N} \sum_{k=0}^{N-1}X(k)e^{j2πkn/N}\)
7. The Nth rot of unity WN is given as ______________
a) ej2πN
b) e-j2πN
c) e-j2π/N
d) ej2π/N
Explanation: We know that the Discrete Fourier transform of a signal x(n) is given as X(k)=\(\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}=\sum_{n=0}^{N-1}x(n)W_N^{kn}\)
Thus we get Nth rot of unity WN=e-j2π/N
8. Which of the following is true regarding the number of computations requires to compute an N-point DFT?
a) N2 complex multiplications and N(N-1) complex additions
b) N2 complex additions and N(N-1) complex multiplications
c) N2 complex multiplications and N(N+1) complex additions
d) N2 complex additions and N(N+1) complex multiplications
Explanation: The formula for calculating N point DFT is given as
X(k)=\(\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}\)
From the formula given at every step of computing we are performing N complex multiplications and N-1 complex additions. So, in a total to perform N-point DFT we perform N2 complex multiplications and N(N-1) complex additions.
9. Which of the following is true?
a) WN*=\(\frac{1}{N} W_N^{-1}\)
b) WN-1=\(\frac{1}{N} W_N*\)
c) WN-1=WN*
d) None of the mentioned
Explanation: If XN represents the N point DFT of the sequence xN in the matrix form, then we know that
XN=WN.xN
By pre-multiplying both sides by WN-1, we get
xN=WN-1.XN
But we know that the inverse DFT of XN is defined as
xN=\(\frac{1}{N} W_N*X_N\)
Thus by comparing the above two equations we get
WN-1 = \(\frac{1}{N} W_N*\)
10. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by ck, then which of the following is true?
a) X(k)=Nck
b) X(k)=ck/N
c) X(k)=N/ck
d) None of the mentioned
Explanation: The Fourier series coefficients are given by the expression
ck=\(\frac{1}{N} \sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}=\frac{1}{N}X(k)\) => X(k)= Nck