(New page: == Computing the Fourier series coefficients for a Discrete Time signal x[n] == === The System === <math>y[n] = x[n] + x[n-1] + x[n-2] + x[n-3]\,</math> === Unit Impulse Response === <m...) |
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== Computing the Fourier series coefficients for a Discrete Time signal x[n] == | == Computing the Fourier series coefficients for a Discrete Time signal x[n] == | ||
| − | + | == The System == | |
<math>y[n] = x[n] + x[n-1] + x[n-2] + x[n-3]\,</math> | <math>y[n] = x[n] + x[n-1] + x[n-2] + x[n-3]\,</math> | ||
| − | + | == Unit Impulse Response == | |
| − | + | ||
<math>x[n] = \delta[n]\,</math> | <math>x[n] = \delta[n]\,</math> | ||
<math>h[n] = \delta[n] + \delta[n-1] + \delta[n-2] + \delta[n-3]\,</math> | <math>h[n] = \delta[n] + \delta[n-1] + \delta[n-2] + \delta[n-3]\,</math> | ||
| + | |||
| + | == Frequency Response == | ||
| + | <math>y[n] = \sum^{\infty}_{\infty} h[n] * x[n] dn\,</math> where <math>x[n] = e^{jwn} \,</math> | ||
| + | |||
| + | <math>y[n] = \sum^{\infty}_{-\infty} (\delta[n] + \delta[n-1]+ \delta[n-2] + \delta[n-3]) e^{jwn} \,</math> | ||
| + | |||
| + | <math>y[n] = \sum^{\infty}_{-\infty} (\delta[m] + \delta[m-1]+ \delta[m-2] + \delta[m-3]) e^{jw(n-m)} \,</math> | ||
| + | |||
| + | <math>y[n] = e^{jwn} \sum^{\infty}_{-\infty} (\delta[m] + \delta[m-1]+ \delta[m-2] + \delta[m-3]) e^{-jwm} \,</math> | ||
| + | |||
| + | <math>H[z] = \sum^{\infty}_{-\infty} (\delta[m] + \delta[m-1]+ \delta[m-2] + \delta[m-3]) e^{-jwm} \,</math> | ||
| + | |||
| + | <math>H[z] = e^{-jw0} + e^{-jw1}+ e^{-jw2}+ e^{-jw3}\,</math> | ||
| + | |||
| + | <math>H[z] = 1 + e^{-jw1}+ e^{-jw2} + e^{-jw3}\,</math> | ||
Revision as of 13:01, 25 September 2008
Contents
Computing the Fourier series coefficients for a Discrete Time signal x[n]
The System
$ y[n] = x[n] + x[n-1] + x[n-2] + x[n-3]\, $
Unit Impulse Response
$ x[n] = \delta[n]\, $
$ h[n] = \delta[n] + \delta[n-1] + \delta[n-2] + \delta[n-3]\, $
Frequency Response
$ y[n] = \sum^{\infty}_{\infty} h[n] * x[n] dn\, $ where $ x[n] = e^{jwn} \, $
$ y[n] = \sum^{\infty}_{-\infty} (\delta[n] + \delta[n-1]+ \delta[n-2] + \delta[n-3]) e^{jwn} \, $
$ y[n] = \sum^{\infty}_{-\infty} (\delta[m] + \delta[m-1]+ \delta[m-2] + \delta[m-3]) e^{jw(n-m)} \, $
$ y[n] = e^{jwn} \sum^{\infty}_{-\infty} (\delta[m] + \delta[m-1]+ \delta[m-2] + \delta[m-3]) e^{-jwm} \, $
$ H[z] = \sum^{\infty}_{-\infty} (\delta[m] + \delta[m-1]+ \delta[m-2] + \delta[m-3]) e^{-jwm} \, $
$ H[z] = e^{-jw0} + e^{-jw1}+ e^{-jw2}+ e^{-jw3}\, $
$ H[z] = 1 + e^{-jw1}+ e^{-jw2} + e^{-jw3}\, $
