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Tuesday, 2 July 2013

Fourier transformation

The convolution integral

  • The output y(t) to an input x(t) is seen as a weighted superposition of impulse response time-shifted by τ.
  • The expression is called the convolution integral and denoted by the same symbol * as in the discrete-time casey(t)=x(τ)h(tτ)dτ=x(t)h(t)

Harmonic signals (complex exponential) are eigenfunctions for LTI systems. That is,
Hf=λf.

Suppose the input is x(t)=Aest. The output of the system with impulse repose h(t) is then
h(tτ)Aesτdτ=h(τ)Aes(tτ)dτ=Aesth(τ)esτdτ=AestH(s)
where H(s) is a scalar dependent only on the parameter s.

Fourier series
Continuous time
X[k]=1TT0x(t)ejkω0tdt

Discrete time
X[k]=1Nn=<N>x[n]ejkΩ0n

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