Next

The Fourier Transform

The Fourier transform of h(t) is given by or which will exist if the energy of the signal h, defined as is finite. In whichcase we may recover the orginal function from H by .

We may consider the application of the Fourier transform to be a function Φ with domain time domain functions and range frequency domain functions. For a time domain function h we will designate its Fourier transform by Φ(h).

Fourier Transform Properties:

  1. Time Scaling Property. If a is a nonzero constant and g(t) = h( at ) for all t, then for each s. We see from this that if the "length" of a signal is decreased while its amplitude is kept constant, then its Fourier transform becomes wider and shorter; ie., the bandwidth increases and the frequency amplitudes decrease. If the length is increased, then its Fourier transform becomes narrower and "taller."
  2. Frequency Scaling Property. If a is a nonzero constant and for all t, then Φ(g)(s)=Φ(h)(as) for each s.
  3. Time Shifting Property. If a is a nonzero constant and g(t) = h( t -a ) for all t, then for each s. Thus the transform of a time shifted function is the transform of the original function multiplied by an exponential factor having a linear phase.
  4. Frequency Shifting Property. If a is a nonzero constant and g(t) = h( t )*ei2πta for all t, then Φ(g)(s) = Φ(h)(s - a) for each s.
  5. Time Convolution Theorem. If for all t, where is the convolution of h and f, then Φ(g)(s) = Φ(h)(s)*Φ(f)(s) for each s. The Fourier transform turns convolution into product. ( is defined to be .)
  6. Frequency Convolution Theorem. If g(t) = h(t)*f(t), for all t, then for each s. The Fourier transform turns product into convolution.
  7. The Correlation Theorem. If g is the correlation of h and f, so, for each t; then Φ(g)(s) = Φ(h)(s)*(Φ(f)(s))* for each s. (z* is the complex comjugate of z: (a+bi)*= a-bi) The correlation of h with itself is called the autocorrelation of h in which case we have Φ(g)(s) = |Φ(h)(s)|2.
  8. Parseval's Theorem. The power of a signal represented by a function h(t) is the same whether computed in the time domain or the frequency domain since .

 

Graphical rule
Please send comments and suggestions to webmaster@theSongEditor.com