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Chapter 1 Introduction > 1.13 Signals correlation - Pg. 40

40 Introduction to CDMA Wireless Communications Output y(k) known Input x(n) ? LTI system h(n) ? Figure 1.26 Blind deconvolution. For k = 0 and assuming h(0) = 0 x(0) = Therefore, the input x(k) is x(k) = 1 y(k) - h(0) k y(0) h(0) h(n) · x(k - n) n=1 for k 1 (1.104) Again the terms in (1.104) do not contain system noise term. In some applications, the system impulse response h(n) is not known and the input x(n) is determined from the measured output y(k) in a process known as blind deconvolution, the concept of which is shown in Figure 1.26. Blind deconvolution is widely used in image signal processing to remove the blurring that degrades the quality of the original image. We will not develop this topic any further as it is beyond the scope of this book. 1.13 Signals correlation Correlation is a measure of the similarity between two signals as one is shifted with respect to the other. The correlation is maxima at the time when the two signals match best. If the two signals are identical, this maximum is when the two copies are synchronous (no delay). Correlation is widely used in applications such as the detection of signals corrupted by channel noise, the estimation of time delay, the time synchronization, pattern matching, and cross spectral analysis. Correlation is equivalent to the time reversed convolution of the two signals. The correlation of two signals is called the cross-correlation and the correlation of the signal with a copy of itself is called the autocorrelation. The average cross correlation function R 12 () of two periodic signals s 1 (t) and s 2 (t), period T 0 is defined as 1 R 12 () = · T 0 - T0 2 s 1 (t) · s 2 (t + ) · dt T0 2 (1.105)