correlation of discrete-time signals transmitted signal, x(n) reflected signal, y(n) = x(n-d) + w(n)...
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Correlation of Discrete-Time Signals
Transmitted Signal, x(n)
Reflected Signal,
y(n) = x(n-D) + w(n)
0 T
Cross-Correlation
• Cross-correlation of x(n) and y(n) is a sequence, rxy(l)
• Reversing the order, ryx(l)
• =>
0, 1, 2,
0, 1, 2,
xyn
xyn
r l x n y n l l
r l x n l y n l
0, 1, 2,
0, 1, 2,
yxn
yxn
r l y n x n l l
r l y n l x n l
xy yxr l r l
Similarity to Convolution
• No folding (time-reversal)
• In Matlab:– Conv(x,fliplr(y))
xyr l x l y l yxr l y l x l
Auto-Correlation
• Correlation of a signal with itself
• Used to differentiate the presence of a like-signal, e.g., zero or one
• Even function
0, 1, 2,xx xxn
r l x n x n l r l l
Properties
• Two sequences, x(n) and y(n), with finite energy
• Find energy of z(n)
z n ax n by n l
2
2 2 2 2
2 2
2
0 0 2 0
zn
n n n
xx yy xy
E ax n by n l
a x n b y n l ab x n y n l
a r b r abr l
Ex Ey
2 2
2
0 0 2 0 ( 0)
0 2 0 0
z xx yy xy
xx xy yy
E a r b r abr l assume b
a ar r l r
b b
Quadratric in (a/b) and positive, discriminant is non-negative:For crosscorrelation case:
For autocorrelation case:
0 0xy xx yy x yr l r r E E
0xx xx xr l r E
Maximum value occurs with zero lag (when signals are perfectly matched)Often normalized to range [-1,1]:
00 0
xy xxxx xx
xxxx yy
r l r ll l
rr r
; 0 1nx n a u n a
2
2
10
1
xxn
n n l l n
n l n l
l
r l x n x n l
a a a a
a la
0
lxxxx
xx
r ll a l
r
Periodic Sequences
• Power signals crosscorrelation:
• Define auto and crosscorrelations over one period of the signals
• If x(n) and y(n) are periodic signals with period N:
• Correlations are also periodic with period N
1lim
2 1
M
xyM
n M
r l x n y n lM
1
0
1 N
xyn
r l x n y n lN
y(n)=x(n)+w(n)
LTI Systems
• Convolution, output of LTI system
• Crosscorrelation between the output and input signal:
• Similarly, input to output is:
k
y n h n x n h k x n k
yx
xx
r l y l x l h l x l x l
h l x l x l
h l r l
xy xxr l h l r l
• Autocorrelation of output:
yy
hh xx
r l y l y l h l x l h l x l
h l h l x l x l
r l r l
0 0 0yy hh xx hh xxk
r r r r k r k
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