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I do not understand this difference equation, so could someone please explain it...

The Fourier Transform gives one answer, but it seems as though there are really more answers. One in paticular, c=0, appears to give better results for the step response...please see attached for explanation of what I am talking about...

$ y[n] - y[n-1] = x[n]\ $
$ \Rightarrow H(\omega) = \frac{1}{1 - e^{-j \omega}} $

From table in book:

$ \begin{align} u[n] &\overset {\mathfrak{F}}{\longleftrightarrow} \frac{1}{1 - e^{-j \omega}} + \sum_{k=-\infty}^{\infty} \pi \delta (\omega - 2 \pi k) \\ 1 &\overset {\mathfrak{F}}{\longleftrightarrow} 2 \pi \sum_{l=-\infty}^{\infty} \delta (\omega - 2 \pi l) \end{align} $
$ \Rightarrow H(\omega) = \frac{1}{1 - e^{-j \omega}} + \sum_{k=-\infty}^{\infty} \pi \delta (\omega - 2 \pi k) - \frac{1}{2} \cdot 2 \pi \sum_{l=-\infty}^{\infty} \delta (\omega - 2 \pi l) $
$ \Rightarrow h[n] = \mathfrak{F}^{-1}(H(\omega)) = u[n] - \frac{1}{2}\ $

Find step response:

$ \begin{align} y[n] &= x[n] * h[n] = u[n] * h[n] = \sum_{k=-\infty}^{\infty} u[k]h[n-k] \\ &= \sum_{k=-\infty}^{\infty} u[k] \left ( u[n-k] - \frac {1}{2} \right ) = \sum_{k=-\infty}^{\infty} u[k]u[n-k] - \frac{1}{2} \sum_{k=-\infty}^{\infty} u[k] \\ &= \sum_{k=0}^{\infty} u[n-k] - \frac{1}{2} \sum_{k=0}^{\infty} 1 = -\infty \end{align} $
(this seems unreasonable!)

Note: The original eq. $ y[n] - y[n-1] = x[n]\ $ can be expressed as:

$ h[n] - h[n-1] = \delta [n]\ $

By observation, $ h[n] = u[n] + c\ $ for any constant c. And further: the step response is divergent for $ c \ne 0\ $.

So, what is c equal to?

once again --mireille.boutin.1, Fri, 26 Oct 2007 14:30:38

Dividing by zero is not recommended.

Please clarify --ross.a.howard.1, Fri, 26 Oct 2007 18:16:00

Can you please clarify where I divided by zero? I still do not know what to do with the arbitrary constant that comes up in the solution...

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva