Question: Channel equalization. We suppose that \(u_{1}, \ldots, u_{m}\) is a signal (time series) that is transmitted (for example by radio). A receiver receives the signal \(y=c * u\), where the \(n\) -vector \(c\) is called the channel impulse response. (See page 138 of the textbook.) In most applications \(n\) is small, e.g., under 10 , and \(m\) is much larger. An equalizer is a \(k\) -vector \(h\) that satisfies \(h * c \approx e_{1}\), the first unit vector of length \(n+k-1\). The receiver equalizes the received signal \(y\) by convolving it with the equalizer to obtain \(z=h * y\).

How are \(z\) (the equalized received signal) and \(u\) (the original transmitted signal) related?

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