![3
Auto correlation measures the similarity of a signal with itself at different
time shifts.
Auto correlation of signal x[n] is defined as:
Rxx(𝑙) = σ𝑛=−∞
∞
𝑥 𝑛 𝑥(𝑛 − 𝑙)
Where 𝑙 = 0, ±1, ± 2……
Auto Correlation](https://image.slidesharecdn.com/autocorrelationcorrected-240906125334-df4399bb/85/Auto-correlation-corrected-on-dsp-pdf-5-320.jpg)
![5
Auto Correlation of the discrete time(DT) signal
Let compute auto correlation of the DT signal x[n] = {− 1, 2, 1}
We know that autocorrelation of x[n]
Rxx(𝑙)= σ𝑛=−∞
∞ 𝑥 𝑛 𝑥(𝑛 − 𝑙)
For 𝑙 = 1;
Rxx(-1)= σ𝑛=−∞
∞ 𝑥 𝑛 𝑥(𝑛 + 1)
= σ𝑛=−∞
∞ [0, −2 , 2]
= 0
𝑙 = 1
= 0
−1 2 1
−1 2 1
−2 2](https://image.slidesharecdn.com/autocorrelationcorrected-240906125334-df4399bb/85/Auto-correlation-corrected-on-dsp-pdf-7-320.jpg)
![For 𝑙 = 2;
Rxx(-2)= σ𝑛=−∞
∞
𝑥 𝑛 𝑥(𝑛 + 2)
= σ𝑛=−∞
∞ [0, 0, −1]
= − 1 𝑙 = 2
= − 1
For 𝑙 = 0;
Rxx(0) = σ𝑛=−∞
∞ 𝑥 𝑛 𝑥(𝑛)
= σ𝑛=−∞
∞ [1, 4, 1]
= 6
6
−1 2 1
−1 2 1
−1](https://image.slidesharecdn.com/autocorrelationcorrected-240906125334-df4399bb/85/Auto-correlation-corrected-on-dsp-pdf-8-320.jpg)
![𝑙 = 0
= 6
For 𝑙 = -1;
Rxx(1)= σ𝑙=−∞
∞
𝑥 𝑛 𝑥(𝑛 − 1)
= σ𝑙=−∞
∞
[−2, 2, 0]
= 0 𝑙 = - 1
= 0
7
−1 2 1
−1 2 1
1 4 1
−1 2 1
−1 2 1
-2 2](https://image.slidesharecdn.com/autocorrelationcorrected-240906125334-df4399bb/85/Auto-correlation-corrected-on-dsp-pdf-9-320.jpg)
![For 𝑙 = -2;
Rxx(2)= σ𝑛=−∞
∞
𝑥 𝑛 𝑥(𝑛 − 2)
= σ𝑛=−∞
∞ [−1, 0, 0]
= −1
𝑙 = -2
= −1
Thus Rxx(𝑙) = {− 1, 0, 6, 0, −1}
8
−1 2 1
−1 2 1
−1](https://image.slidesharecdn.com/autocorrelationcorrected-240906125334-df4399bb/85/Auto-correlation-corrected-on-dsp-pdf-10-320.jpg)
![9
clc;
close all;
clear all;
x = [-1, 2, 1]
t = 0:length(x)-1;
subplot(2,1,1);
stem(t, x,'linewidth',2);
auto_corr = xcorr(x);
lag = -length(x)+1:length(x)-1; % Lag values for xcorr resul
subplot(2,1,2);
stem(lag, auto_corr,'linewidth',2 );
disp('Rxx:');
disp(auto_corr);
Auto Correlation Using MATLAB](https://image.slidesharecdn.com/autocorrelationcorrected-240906125334-df4399bb/85/Auto-correlation-corrected-on-dsp-pdf-11-320.jpg)
Auto correlation(corrected) on dsp. .pdf
- 2. Saroar Hossain ID: 2002221 Md.Mahbub Hasan ID:2002271 Presented By Presented to Prof. Dr. Md. Zamil Sultan Chairman Dept. of EEE Rimu Rahman Munna ID: 2002267 Rakibul Hasan ID:2002280
- 3. AUTO CORRELATION IN DIGITAL SIGNAL PROCESSING 1
- 4. 2 In digital signal processing (DSP), correlation is a fundamental operation used to measure the similarity between two signals. Correlation Types of Correlation There are two main types of correlation in DSP 1. Auto Correlation 2. Cross Correlation
- 5. 3 Auto correlation measures the similarity of a signal with itself at different time shifts. Auto correlation of signal x[n] is defined as: Rxx(𝑙) = σ𝑛=−∞ ∞ 𝑥 𝑛 𝑥(𝑛 − 𝑙) Where 𝑙 = 0, ±1, ± 2…… Auto Correlation
- 6. 4 Auto Correlation 0 0.5 1 1.5 2 2.5 0 1 2 3 4 Transmitted Signal 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 Received Signal 𝑥 𝑛 𝑥 𝑛 − 𝑙
- 7. 5 Auto Correlation of the discrete time(DT) signal Let compute auto correlation of the DT signal x[n] = {− 1, 2, 1} We know that autocorrelation of x[n] Rxx(𝑙)= σ𝑛=−∞ ∞ 𝑥 𝑛 𝑥(𝑛 − 𝑙) For 𝑙 = 1; Rxx(-1)= σ𝑛=−∞ ∞ 𝑥 𝑛 𝑥(𝑛 + 1) = σ𝑛=−∞ ∞ [0, −2 , 2] = 0 𝑙 = 1 = 0 −1 2 1 −1 2 1 −2 2
- 8. For 𝑙 = 2; Rxx(-2)= σ𝑛=−∞ ∞ 𝑥 𝑛 𝑥(𝑛 + 2) = σ𝑛=−∞ ∞ [0, 0, −1] = − 1 𝑙 = 2 = − 1 For 𝑙 = 0; Rxx(0) = σ𝑛=−∞ ∞ 𝑥 𝑛 𝑥(𝑛) = σ𝑛=−∞ ∞ [1, 4, 1] = 6 6 −1 2 1 −1 2 1 −1
- 9. 𝑙 = 0 = 6 For 𝑙 = -1; Rxx(1)= σ𝑙=−∞ ∞ 𝑥 𝑛 𝑥(𝑛 − 1) = σ𝑙=−∞ ∞ [−2, 2, 0] = 0 𝑙 = - 1 = 0 7 −1 2 1 −1 2 1 1 4 1 −1 2 1 −1 2 1 -2 2
- 10. For 𝑙 = -2; Rxx(2)= σ𝑛=−∞ ∞ 𝑥 𝑛 𝑥(𝑛 − 2) = σ𝑛=−∞ ∞ [−1, 0, 0] = −1 𝑙 = -2 = −1 Thus Rxx(𝑙) = {− 1, 0, 6, 0, −1} 8 −1 2 1 −1 2 1 −1
- 11. 9 clc; close all; clear all; x = [-1, 2, 1] t = 0:length(x)-1; subplot(2,1,1); stem(t, x,'linewidth',2); auto_corr = xcorr(x); lag = -length(x)+1:length(x)-1; % Lag values for xcorr resul subplot(2,1,2); stem(lag, auto_corr,'linewidth',2 ); disp('Rxx:'); disp(auto_corr); Auto Correlation Using MATLAB
- 12. 10
- 13. 11 • The maximum value of the auto-correlated signal is obtained when time-shift is zero that is at 𝑙 = 0. Because at 𝑙 = 0 the overlapping signal best matches the given signal. • Rxx(𝑙) is a symmetric function, that is Rxx(𝑙) = Rxx(- 𝑙 ) Important Feature of Auto Correlation
- 14. 12 • Auto-correlation is used to identify repeating patterns within a signal • Speech processing • Signal synchronization in communication systems • Radar system • Weather forecasting • Security system Applications of Auto Correlation
- 15. THANK YOU