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I am following the book: Blind equalization and Identification by Zhi Ding and Ye Li. In Chapter 2, the concept of T spaced equalizers is presented. It is mentioned that the output of the channel is $x[k] = \sum_{i=-\infty}^\infty h[i]s[k-i] + w[k]$ where $h$ is the impulse response, $s$ are the symbols / source input and $w[k]$ is iid noise. Then it is defined that *

"When the channel is non-deal, its impulse response $h[k]$ has more than one nonzero component. Consequently, undesirable signal distrotion as the channel output $x[k]$ depends on multiple symbols in $\{s[k]\}$. This phenomenon is known as inter-symbol interference (ISI)."

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I do not understand why the impulse response has to be zero when it is ideal channel - no ISI. If $h$ is zero then $h[i].s[k-i] = 0$ where $dot$ is the multiplication operator. Or am I mistaken and in the equation $h$ is convolved with $s$ and not multiplied and essentially impulse response will be zero for ideal channel. What is the concept and reason? I do know that convolution in time domain is equivalent to multiplication in frequency domain. But the above representation is in time domain. A brief explanation will work and then I can follow on from the key points mentioned in the answer. Thank you.

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The book doesn't say that the impulse response must be zero for an ideal channel. It says that an ideal channel has exactly one, and not more than one, non-zero component, i.e. the ideal channel's impulse response is an impulse, which means that the signal is only delayed but not distorted.

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  • $\begingroup$ This is correct, except that the signal can also be attenuated or amplified, in addition to delayed. $\endgroup$ – MBaz Oct 30 '15 at 21:20
  • $\begingroup$ @MBaz: That could be, it all depends on the definition of 'distortionless'. I believe to remember that some sources define it as a pure delay, with a gain of unity. Anyway, good you pointed it out. $\endgroup$ – Matt L. Oct 30 '15 at 21:24
  • $\begingroup$ Yes, the strict definition of 'distortionless' is what you say. Since all physical channels attenuate the input anyway, in communications 'distortion' is usually reserved to describe a modification to the signal's shape beyond a constant gain. $\endgroup$ – MBaz Oct 30 '15 at 21:50
  • $\begingroup$ ISI is caused by smearing out of the data symbols over time by a dispersive channel, i.e. a channel with an impulse response that is not a single impulse. The longer the channel's impulse response, the more severe the ISI will be on average. In the time/sample domain, the signal is always convolved with the impulse response, that's how linear time-invariant systems work. You get multiplication in the frequency domain. $\endgroup$ – Matt L. Nov 2 '15 at 8:26

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