Solutions Manual Digital Signal Processing System Analysis and Design 2nd Edition By Paulo S. R. Diniz

Solutions Manual
Digital Signal Processing System Analysis and Design 2nd Edition
By
Paulo S. R. Diniz,
Eduardo A. B. da Silva,
Sergio L. Netto


( All Chapters Included - 100% Verified Solutions )




1

,Chapter 1

D ISCRETE - TIME SYSTEMS

1.1 (a) y(n) = (n + a)2 x(n + 4)
• Linearity:
2
H{kx (n)} = (n + a) kx (n + 4) , k ∈ R
2
= k (n + a) x (n + 4)
= kH{x (n)}
2
H{x1 (n) + x2 (n)} = (n + a) [x1 (n + 4) + x2 (n + 4)]
2 2
= (n + a) x1 (n + 4) + (n + a) x2 (n + 4)
= H{x1 (n)} + H{x2 (n)}


and therefore the system is linear.
• Time Invariance:
2
y (n − n0 ) = (n − n0 + a) x (n − n0 + 4)
2
H{x (n − n0 )} = (n + a) x (n − n0 + 4)
then y (n − n0 ) 6= H{x (n − n0 )}, and the system is time varying.
• Causality:
2
H{x1 (n)} = (n + a) x1 (n + 4)
2
H{x2 (n)} = (n + a) x2 (n + 4)
If x1 (n) = x2 (n) for n < n0 , and x1 (n0 + 3) 6= x2 (n0 +3), then for n = n0 − 1 < n0 :
2
H{x1 (n)} = (n0 − 1 + a) x1 (n0 + 3)
n=n0 −1
2
H{x2 (n)} = (n0 − 1 + a) x2 (n0 + 3)
n=n0 −1

and then H{x1 (n)} =
6 H{x2 (n)} for n < n0 . Thus, the system is noncausal.

(b) y(n) = ax(n + 1)
• Linearity:
H{kx (n)} = akx (n + 1) , k ∈ R
= kH{x (n)}
H{x1 (n) + x2 (n)} = a [x1 (n + 1) + x2 (n + 1)]
= ax1 (n + 1) + ax2 (n + 1)
= H{x1 (n)} + H{x2 (n)}

1


2

,2 CHAPTER 1. DISCRETE-TIME SYSTEMS


and therefore the system is linear.
• Time Invariance:
y (n − n0 ) = ax [(n − n0 ) + 1]
H{x (n − n0 )} = ax [(n − n0 ) + 1]
then y (n − n0 ) = H{x (n − n0 )}, and the system is time invariant.
• Causality:
H{x1 (n)} = ax1 (n + 1)
H{x2 (n)} = ax2 (n + 1)
therefore, if x1 (n) = x2 (n) for n < n0 , and x1 (n0 ) 6= x2 (n0 ), then for n = n0 − 1 < n0 :

H{x1 (n)} = ax1 (n0 − 1 + 1) = ax1 (n0 )
n=n0 −1

H{x2 (n)} = ax2 (n0 − 1 + 1) = ax2 (n0 )
n=n0 −1

and then H{x1 (n)} =
6 H{x2 (n)} for n < n0 . Thus, the system is noncausal.

(c) y(n) = x(n + 1) + x3 (n − 1)
• Linearity:
H{kx (n)} = kx (n + 1) + k 3 x3 (n − 1) , k ∈ R
kH{x (n)} = kx (n + 1) + kx3 (n − 1)
H{kx (n)} 6= kH{x (n)}
and therefore the system is nonlinear.
• Time Invariance:
y (n − n0 ) = x (n − n0 + 1) + x3 (n − n0 − 1)
H{x (n − n0 )} = x (n − n0 + 1) + x3 (n − n0 − 1)
then y (n − n0 ) = H{x (n − n0 )}, and the system is time invariant.
• Causality:
H{x1 (n)} = x1 (n + 1) + x31 (n − 1)
H{x2 (n)} = x2 (n + 1) + x32 (n − 1)
If x1 (n) = x2 (n) for n < n0 , and x1 (n0 ) 6= x2 (n0 ), then, for n = n0 − 1 < n0 :

H{x1 (n)} = x1 (n0 − 1 + 1) + x31 (n0 − 1 − 1) = x1 (n0 ) + x31 (n0 − 2)
n=n0 −1

H{x2 (n)} = x2 (n0 − 1 + 1) + x32 (n0 − 1 − 1) = x2 (n0 ) + x32 (n0 − 2)
n=n0 −1

and then H{x1 (n)} =
6 H{x2 (n)} for n < n0 . Thus, the system is noncausal.

(d) y(n) = x(n) sin(ωn)
• Linearity:
H{kx (n)} = kx (n) sin (ωn) , k ∈ R
= kH{x (n)}
H{x1 (n) + x2 (n)} = [x1 (n) + x2 (n)] sin (ωn)
= x1 (n) sin (ωn) + x2 (n) sin (ωn)
= H{x1 (n)} + H{x2 (n)}
and therefore the system is linear.




3

, 3


• Time Invariance:
y (n − n0 ) = x (n − n0 ) sin (ωn − ωn0 )
H{x (n − n0 )} = x (n − n0 ) sin (ωn)
then y (n − n0 ) 6= H{x (n − n0 )}, and the system is time varying.
• Causality:
H{x1 (n)} = x1 (n) sin (ωn)
H{x2 (n)} = x2 (n) sin (ωn)
therefore if x1 (n) = x2 (n) for n < n0 , then H{x1 (n)} = H{x2 (n)} for n < n0 . Thus, the
system is causal.

(e) y(n) = x(n) + sin(ωn)
• Linearity:
H{kx (n)} = kx (n) + sin (ωn) , k ∈ R
kH{x (n)} = k [x (n) + sin (ωn)]
= kx (n) + k sin (ωn)
H{kx (n)} 6= kH{x (n)}
and therefore the system is nonlinear.
• Time Invariance:
y (n − n0 ) = x (n − n0 ) + sin (ωn − ωn0 )
H{x (n − n0 )} = x (n − n0 ) + sin (ωn)
then y (n − n0 ) 6= H{x (n − n0 )}, and the system is time varying.
• Causality:
H{x1 (n)} = x1 (n) + sin (ωn)
H{x2 (n)} = x2 (n) + sin (ωn)
If x1 (n) = x2 (n) for n < n0 , then H{x1 (n)} = H{x2 (n)} for n < n0 . Thus, the system is
causal.

x(n)
(f) y(n) = x(n+3)
• Linearity:
kx (n)
H{kx (n)} = ,k ∈ R
kx (n + 3)
x (n)
=
x (n + 3)
x (n)
kH{x (n)} = k
x (n + 3)
H{kx (n)} 6= kH{x (n)}
and therefore the system is nonlinear.
• Time Invariance:
x (n − n0 )
y (n − n0 ) =
x (n − n0 + 3)
x (n − n0 )
H{x (n − n0 )} =
x (n − n0 + 3)
then y (n − n0 ) = H{x (n − n0 )}, and the system is time invariant.




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