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# Fourier transform of typical signals

• Impulse

As shown above, • Unit Step

As shown above, • Constant

As shown above, This is a useful formula.

• Complex exponential

The spectrum of a complex exponential can be found from the above due to the frequency shift property: • Sinusoids  Similarly, we have • Exponential decay - right-sided      • Exponential decay - left-sided Due to the time reversal property, we also have (for ): or • Exponential decay - two-sided As the two-sided exponential decay is the sum of the right and left-sided exponential decays, its spectrum of is the sum of their spectra due to linearity: • Comb function

The comb function is defined as Its Fourier series coefficient is: and its spectrum is: We see that the spectrum of an impulse train with time interval is also an impulse train with frequency interval . Also, according to the definition of the Fourier transform, we have     Therefore we have this equation which can be compared with the equation in continuous case: • Square wave

A square wave or rectangular function of width can be considered as the difference between two unit step functions and due to linearity, its Fourier spectrum is the difference between the two corresponding spectra:     • Sinc function

The spectrum of an ideal low-pass filter is and its impulse response can be found by inverse Fourier transform: • Triangle function As is an even function, its Fourier transform is         Alternatively, as the triangle function is the convolution of two square functions ( ), its Fourier transform can be more conveniently obtained according to the convolution theorem as: • Gaussian function

The Fourier transform of a Gaussian or bell-shaped function is       Here we have used the identity We see that the Fourier transform of a bell-shaped function is also a bell-shaped function: Note that the area underneath either or is unity. Moreover, due to the property of time and frequency scaling, we have: (Note that if , then above is a normal distribution with variance and mean .) If we let , becomes narrower and taller and approaches , and its spectrum becomes wider and approaches constant . On the other hand, if we rewrite the above as and let , approaches 1 and approaches .   Next: Examples Up: handout3 Previous: Properties of Fourier Transform
Ruye Wang 2009-07-05