Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 257

Tutorial 23

Q: Find the autocorrelation function of x ð t Þ¼ Q 1 ð t 0 : 5 Þ:
Note that x(t)isadeterministic signal.
Solution:FromTables:R(s) = $ - ? x(k)x(s + k)dk.NotethatR(s) = x(s)*x(-s).
We follow the same steps that are necessary to find the convolution of two signals.

From Tables: Q T ð t Þ¼ 1 ; T = 2 t T = 2

0 ; elsewhere

¼ 1 ;

x ð k Þ¼ Y 1 ð k 0 : 5 Þ¼ 1 ; 0 : 5 k 0 : 5 0 : 5

0 k 1

)

0 ; elsewhere 0 ; elsewhere

8
<
:

9
=
;

¼

1 ; s

k 1 s

|{z}
start

| {z }
end x ð s þ k Þ¼ 1 ;

0 s þ k 1

)

0 ; elsewhere

0 ; elsewhere

Note that since we have x(s + k) inside the integral, a positive k-shift s will give
negative shift to the function w.r.t the y-axis (contrary to the case of convolution).
Now we move the function x(s + k) from left to right while x(k) is fixed. We get
the following cases:

Π 1 ( λ )

1

λ

−1 0 1

x ( λ ) = Π 1 ( λ − 0.5)

1

λ

−1 0 1

x ( τ + λ ) [ τ > 0 ]

1

λ

−1 0 1

− τ 1 − τ

(Start) (End)

R ( τ )

1

τ

−1 0 1

Case 1: 1 s\0 ð i.e. ; s [ 1 Þ) R ð s Þ¼ 0 [no overlap].

0 1 s\1 ð i.e. ; 0\s 1 Þ) R ð s Þ¼ R 1 s

Case 2:

0 dk ¼ 1 s :

Case 3: 0 s 1 ð i.e. ; 1 s 0 Þ) R ð s Þ¼ R 1 s dk ¼ 1 þ s.
Case 4: s [ 1 ð i.e. ; s\ 1 Þ) R ð s Þ¼ 0 [no overlap].