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− | DTFT | + | == Discrete Time Fourier Transform (DTFT) with example == |
+ | |||
+ | a Slecture by ECE Student Fabian Faes | ||
+ | |||
+ | Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin. | ||
+ | |||
+ | ---- | ||
+ | |||
+ | ---- | ||
+ | |||
+ | == Definition == | ||
+ | |||
+ | The discrete time fourier transform (DTFT) of a finite energy aperiodic signal x[n] can be given by the equation listed below. IT is a representation in terms of a complex exponential sequence <span class="texhtml">''e''<sup>''j''ω''n''</sup></span>, where <span class="texhtml">ω</span> is a real frequency variable. | ||
+ | |||
+ | <math>X(e^{j{\omega}}) = X({\omega}) = {\sum_{n = -{\infty}}^{\infty} x[n]e^{-j{\omega}n}}</math> | ||
+ | |||
+ | <span class="texhtml">''X''(ω)</span> can also be represented in terms of its magnitude and phase as shown below: | ||
+ | |||
+ | '''magnitude''' | ||
+ | |||
+ | <span class="texhtml">''X''(ω) = | ''X''(ω) | ''e''<sup>''j''θω</sup></span> | ||
+ | |||
+ | '''phase''' | ||
+ | |||
+ | <math>{\theta}({\omega}) = {\angle}X({\omega})</math> | ||
+ | |||
+ | |||
+ | ---- | ||
+ | |||
+ | |||
+ | == Periodicity Property == | ||
+ | |||
+ | We note that <math>X({\omega})</math> is periodic with period <math>2{\pi}</math> since | ||
+ | |||
+ | <math>X({\omega} + 2{\pi}) = {\sum_{n = -{\infty}}^{\infty} x[n]e^{-j({\omega} + 2{\pi})n}}</math> | ||
+ | |||
+ | <math>X({\omega} + 2{\pi}) = {\sum_{n = -{\infty}}^{\infty} x[n]e^{-j{\omega}n}e^{-j2{\pi}n}}</math> | ||
+ | |||
+ | <math>X({\omega} + 2{\pi}) = {\sum_{n = -{\infty}}^{\infty} x[n]e^{-j{\omega}n}}</math> | ||
+ | |||
+ | <math>X({\omega} + 2{\pi}) = X({\omega})</math> | ||
+ | |||
+ | for clarification <math>e^{-j2{\pi}n} = 1</math> since | ||
+ | |||
+ | <math>e^{-j2{\pi}n} = cos(2{\pi}) + jsin(2{\pi}n)</math> | ||
+ | |||
+ | <math>e^{-j2{\pi}n} = 1 + j0</math> | ||
+ | |||
+ | <math>e^{-j2{\pi}n} = 1</math> | ||
+ | |||
+ | However it is interesting to note that this does not apply if we are dealing with a continous signal. This is a common mistake made by students. | ||
+ | |||
+ | <math>e^{-j2{\pi}n} = (e^{-j2{\pi}})^n = 1^n = 1</math> | ||
+ | |||
+ | <math>e^{-j2{\pi}t} = (e^{-j2{\pi}})^t = 1^t {\neq} 1</math> | ||
+ | |||
+ | This is because t can lead to a <math>{\pm}</math> solution due to t not being a definite value such as n. | ||
+ | |||
+ | |||
+ | ---- | ||
+ | |||
+ | == Complex Exponential Example == |
Revision as of 22:59, 30 September 2014
Contents
Discrete Time Fourier Transform (DTFT) with example
a Slecture by ECE Student Fabian Faes
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
Definition
The discrete time fourier transform (DTFT) of a finite energy aperiodic signal x[n] can be given by the equation listed below. IT is a representation in terms of a complex exponential sequence ejωn, where ω is a real frequency variable.
$ X(e^{j{\omega}}) = X({\omega}) = {\sum_{n = -{\infty}}^{\infty} x[n]e^{-j{\omega}n}} $
X(ω) can also be represented in terms of its magnitude and phase as shown below:
magnitude
X(ω) = | X(ω) | ejθω
phase
$ {\theta}({\omega}) = {\angle}X({\omega}) $
Periodicity Property
We note that $ X({\omega}) $ is periodic with period $ 2{\pi} $ since
$ X({\omega} + 2{\pi}) = {\sum_{n = -{\infty}}^{\infty} x[n]e^{-j({\omega} + 2{\pi})n}} $
$ X({\omega} + 2{\pi}) = {\sum_{n = -{\infty}}^{\infty} x[n]e^{-j{\omega}n}e^{-j2{\pi}n}} $
$ X({\omega} + 2{\pi}) = {\sum_{n = -{\infty}}^{\infty} x[n]e^{-j{\omega}n}} $
$ X({\omega} + 2{\pi}) = X({\omega}) $
for clarification $ e^{-j2{\pi}n} = 1 $ since
$ e^{-j2{\pi}n} = cos(2{\pi}) + jsin(2{\pi}n) $
$ e^{-j2{\pi}n} = 1 + j0 $
$ e^{-j2{\pi}n} = 1 $
However it is interesting to note that this does not apply if we are dealing with a continous signal. This is a common mistake made by students.
$ e^{-j2{\pi}n} = (e^{-j2{\pi}})^n = 1^n = 1 $
$ e^{-j2{\pi}t} = (e^{-j2{\pi}})^t = 1^t {\neq} 1 $
This is because t can lead to a $ {\pm} $ solution due to t not being a definite value such as n.