Frequency-Domain Analysis - Introduction to Digital Signal Processing and Filter Design - page 167

f p (3 − m)

− 10 − 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 9 10 11 12

( f )

f(4 − m)

− 11 − 10 − 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 0123456789 10 1 12 m

( g )

f p (5 − m)

− 10 − 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 0123456789 10 1 12 m

( h )

y(n)

n

− 12 − 11 − 10 − 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 0123456789 10 1 12 13

( i )

Figure 3.31 ( Continued )

But we notice that this does not match the result of the linear convolution
y 1 (n)

[1 . 01 . 61 . 61 . 36 0 . 60 . 24]. It is obvious that the
length of y 1 (n) is 6, whereas x p (n), f p (n) , X p (k), F p (k) ,and y p (n) are all of
length 4, and for that reason alone, we do not expect the two results to match.
If we look carefully at Figure 3.31 and Figure 3.30, we see another reason why
they do not match. In Figures 3.31, f( 4 −

= x(n) ∗ f(n) =

m), f ( 5 − m) , f( 6 − m), f ( 7 − m)

are found to be the same as the sequence f( 0 ) , f( 1 − m) , f( 2 − m) , f( 3 − m) ,