- 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(t−τ)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(t−τ)Aesτdτ=∫∞−∞h(τ)Aes(t−τ)dτ=Aest∫∞−∞h(τ)e−sτdτ=AestH(s)
where H(s) is a scalar dependent only on the parameter s.
Fourier series
Continuous time
X[k]=1T∫T0x(t)e−jkω0tdt
Discrete time
X[k]=1N∑n=<N>x[n]e−jkΩ0n
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