Signal processing

1. Let $$x\left[ n \right] = {\left( { – {1 \over 9}} \right)^n}u\left( n \right) – {\left( { – {1 \over 3}} \right)^n}u\left( { – n – 1} \right).$$ The Region of Convergence (ROC) of the z-transform of x[n]

”Is
”Is
”Does

Detailed SolutionLet $$x\left[ n \right] = {\left( { – {1 \over 9}} \right)^n}u\left( n \right) – {\left( { – {1 \over 3}} \right)^n}u\left( { – n – 1} \right).$$ The Region of Convergence (ROC) of the z-transform of x[n]

3. The voltage across an impedance in a network is V(s) = Z(s). I(s), where V(s), Z(s) and I(s) are the Laplace transform of the corresponding time functions v(t), z(t) and i(t). The voltage v(t) is

v(t) = z(t).i(t)
$$vleft( t ight) = intlimits_0^t {ileft( au ight)} zleft( {t - au } ight)d au $$
$$vleft( t ight) = intlimits_0^t {ileft( au ight)} zleft( {t + au } ight)d au $$
v(t) = z(t) + i(t)

Detailed SolutionThe voltage across an impedance in a network is V(s) = Z(s). I(s), where V(s), Z(s) and I(s) are the Laplace transform of the corresponding time functions v(t), z(t) and i(t). The voltage v(t) is

6. A periodic signal x(t) has a trigonometric Fourier series expansion $$x\left( t \right) = {a_0} + \sum\limits_{n = 1}^\infty {\left( {{a_n}\,\cos \,n{\omega _0}t + {b_n}\sin \,n{\omega _0}t} \right)} $$ If $$x\left( t \right) = – x\left( { – t} \right) = – x\left( {{{t – \pi } \over {{\omega _0}}}} \right),$$ we can conclude that

an are zero for all n and bn are zero for n even
an are zero for all n and bn are zero for n odd
an are zero for n even and bn are zero for n odd
an are zero for n odd and bn are zero for n even

Detailed SolutionA periodic signal x(t) has a trigonometric Fourier series expansion $$x\left( t \right) = {a_0} + \sum\limits_{n = 1}^\infty {\left( {{a_n}\,\cos \,n{\omega _0}t + {b_n}\sin \,n{\omega _0}t} \right)} $$ If $$x\left( t \right) = – x\left( { – t} \right) = – x\left( {{{t – \pi } \over {{\omega _0}}}} \right),$$ we can conclude that

7. The magnitude and phase of the complex Fourier series coefficient ak of a periodic signal x(t) are shown in the figure. Choose the correct statement from the four choices given. Notation: C is the set of complex number, R is the set of purely real numbers, and P is the set of purely imaginary numbers.

$$xleft( t ight) in R$$
$$xleft( t ight) in P$$
$$xleft( t ight) in left( {C - R} ight)$$
The information given is not sufficient to draw any conclusion about x(t)

Detailed SolutionThe magnitude and phase of the complex Fourier series coefficient ak of a periodic signal x(t) are shown in the figure. Choose the correct statement from the four choices given. Notation: C is the set of complex number, R is the set of purely real numbers, and P is the set of purely imaginary numbers.

8. For an N-point FFT algorithm with N = 2m, which one of the following statements is TRUE?

It is not possible to construct a signal flow graph with both input and output in normal order
The number of butterflies in the mn state is $$ rac{N}{m}$$
In-place computation requires storage of only 2N node data
Computation of a butterfly requires only one complex multiplication

Detailed SolutionFor an N-point FFT algorithm with N = 2m, which one of the following statements is TRUE?

9. A discrete time linear shift-invariant system has an impulse response h[n] with h[0] = 1, h[1] = -1, h[2] = 2, and zero otherwise. The system is given an input sequence x[n] with x[0] = x[2] = 1 and zero otherwise. The number of nonzero samples in the output sequence y[n], and the value of y[2] are, respectively

5, 2
6, 2
6, 1
5, 3

Detailed SolutionA discrete time linear shift-invariant system has an impulse response h[n] with h[0] = 1, h[1] = -1, h[2] = 2, and zero otherwise. The system is given an input sequence x[n] with x[0] = x[2] = 1 and zero otherwise. The number of nonzero samples in the output sequence y[n], and the value of y[2] are, respectively

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