Frequency-Domain Analysis - Introduction to Digital Signal Processing and Filter Design - page 133

ae − jω ) e jω /(e jω This infinite series converges to 1 /( 1 − = − a) when

ae − jω < 1, that is, when |

< 1. So, the DTFT of ( 0 . 4 ) n u(n) is 1 /( 1 0 . 4 e − jω ) a | −

− 0 . 4 ) n u(n) is 1 /( 1 + 0 . 4 e − jω ) . Note that both of them are causal and the DTFT of (

sequences.
If we are given a sequence x 13 (n)

a | n | ,where |

< 1, we split the sequence
as a causal sequence x 1 (n) from 0 to ∞ , and a noncausal sequence x 3 (n)
from −∞ to − 1. In other words, we can express x 1 (n)

= a |

a n u(n) and x 3 (n) = =

a − n u( − n − 1 ) .WederivetheDTFTof x 13 (n) as

∞ − 1

X 13 (e jω ) a n e − jωn a − n e − jωn X 1 (e jω ) X 3 (e jω ) = + +

n =−∞

n = 0

n in the second summation for X 3 (e jω ) ,weget Substituting m =−

∞ ∞

ae − jω ae jω

n m

X 13 (e jω )

= +

n = 0 m = 1

∞ ∞

ae − jω n ae jω m

= − 1 +

n = 0 m = 0

1 1

=

ae − jω −

1 + for | a | < 1

ae jω 1 − 1 −

ae jω
1 −

1

=

ae − jω + 1 − ae jω

a 2
1 − 2 a cos ω

1 −

= for | a | < 1

+ a 2

Hence we have shown that

a 2 1 −

a | n | ⇔ for | a | < 1

−

2 a cos ω

+ a 2

1

< 1. From the result a n u(n) ae − jω ) , These results are valid when | a | ⇔

1 /( 1

−

by application of the time-reversal property, we also find that x 4 (n) = x 1 ( − n) =

a − n u( ae jω ) for − n) ⇔ 1 /( 1 − | a | < 1 whereas we have already determined that

ae jω ) . Note that x 3 (n) is obtained from
x 4 (n) by deleting the sample of x 4 (n) at n

a − n u( ae jω /( 1

x 3 (n) = − n − 1 ) ⇔ −

x 3 (n) .
We used this result in deriving X 3 (e jω ) above. The sequence x 13 (n) is plotted in
Figure 3.13, while the plots of x 1 (n), x 3 (n) are shown in Figures 3.14 and 3.15,
respectively.

= 0, specifically, x 4 (n) − 1 =