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To answer your second question: To determine the cross correlation with discrete tones you can get the frequency response of your correlation by treating the sequence as the time reversed coefficients of an FIR filter (since the FIR filter performs convolution of your signal with the coefficients, and correlation is convolution with one of the sequences time ...

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I assume that this is a discrete-time problem where the maximum-length sequence is a pseudorandom sequence $x[k]$ of $\pm 1$ values and the noise is a sequence $n[k]$ of independent identically distributed (iid) zero-mean random variables with variance $\sigma^2$. Then, $\sum_{k=0}^{N-1} x[k]n[k]$ is also a sum of $N$ iid random variables and its variance ...

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The correlation is used to find the time shift between signsls while convolution represents system response to predefined input. Since the system response depends on previous input, rether then future input, the sign of the time distance is negative.

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You are right. In the solution, the second line which performs the long division $$H(z) = -4 + \frac{ 5 + \frac{7}{2} z^{-1} }{1 - \frac{3}{4}z^{-1} + \frac{1}{8} z^{-1} }$$ is wrong and should be corrected as: $$H(z) = -4 + \frac{ 5 - 3 z^{-1} }{1 - \frac{3}{4}z^{-1} + \frac{1}{8} z^{-1} } .$$ However, the partial fraction expansion at the following ...

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This is a homework type problem, so I'll give you a few hints to help you solve it yourself (and learn something while doing so). From what is given, we can assume that the given difference equation describes a causal discrete-time system. Let $h[n]$ denote the impulse response. It must satisfy h[n]=h[n-1]+\frac{1}{N}\big(\delta[n]-\delta[n-N]\big),\...

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