Values of xp n and hp n outside the range 0 N 1 are generated by periodic

Values of xp n and hp n outside the range 0 n 1 are

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Values of x p [- n ] and h p [ n ] outside the range (0, N - 1) are generated by periodic extension. One way to visualize the process is to line up x [ k ] clockwise around a circle & h [ k ] counterclockwise (folded).
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51 Example 2.7 Find the periodic convolution of and with the period of N = 3 using the cyclic method 3} 2, , 1 { n x p 2} 0, , 1 { n h p
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52 Solution Rotate outer sequence (folded h ) clockwise Rotate outer sequence (folded h ) clockwise 5} 8, , 5 { n y
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53 Hardware implementation of convolution Need memory, adder and multiplier Perform operation of addition, multiplication and shifting
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54
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55 Discrete Correlation Correlation is a measure of similarity between two signals and is found using a process similar to convolution. Correlation is the convolution of one signal with a folded version of the other. The discrete cross-correlation (denoted  ) of x [ n ] and h [ n ] is defined by: k h n k x n k h k x n h n x n r k k xh   k x n k h n k x k h n x n h n r k k hx  
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56 To find r xh [ n ], the last element of h [ n ] is lined up with the first element of x [ n ] & start shifting h [ n ] past x [ n ]. The pointwise product of the overlapping values are summed up to generate the correlation. This is equivalent to performing the convolution of x [ n ] & the folded signal h [- n ] . The starting index of the correlation equals the sum of the starting indices of x [ n ] and h [- n ] . Similarly, r hx [ n ] equals the convolution of x [- n ] & h [ n ], & its starting index equals the sum of the starting indices of x [- n ] & h [ n ] .
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57 However, r xh [ n ] does not equal to r hx [ n ] . The two are folded versions of each other & related by r xh [ n ] = r hx [- n ] . Some equations need to be remembered: Correlation length : N x + N h - 1 Correlation sum : n h n x n h n x n r xh n x n h n x n h n r hx n h n x n r
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58 Autocorrelation The correlation r xx [ n ] of a signal x [ n ] with itself is called the autocorrelation. It is an even symmetric function ( r xx [ n ] = r xx [- n ]) with a maximum at n = 0 and satisfies the inequality . Correlation is an effective method of detecting signals buried in noise. Noise is essentially uncorrelated with the signal
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59 It means that if we correlate a noisy signal with itself, the correlation will be due only to the signal (if present) and will exhibit a sharp peak at n = 0. The autocorrelation is always even symmetric with a maximum at the origin. Some equations need to be remembered: n x n x n x n x n r xx n r n r xx xx 0 xx xx r n r
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60 Question
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61 Answer
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62 Answer
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63 Example 2.8 Given x [ n ] = a n u [ n ], | a | < 1. Find r xx [ n ] for n ≥ 0
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64 Solution Since x [ k - n ] = a k-n u [ k - n ] starts at k = n , then Since autocorrelation is an even symmetric function, we have  n k m m n m n m n m n k k k xx a a a a a a a a n k x k x n r 0 0 2 2 1 1 2 1 a a n r n xx
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65 No man an ever reached to excellence in any ONE art or profession without having passed through the slow and painful process of study and preparation.”
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