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Remember Convolution
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Claim: The impulse response of a system (transform)
completely describes it y[t] = T[x[t]] y[t] = T[ Σ x[k]δ[t-k] ] y[t] = Σ T[ x[k]δ[t-k] ] y[t] = Σ x[k] T[δ[t-k]] y[t] = Σ x[k] h[t-k] This is called convolution and not only does it describe transforms. It is handy elsewhere. Often denoted: f(x)*g(x) Illustration |
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Fourier Is The Man
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Joseph
Fourier Any complex wave can be broken down into sine waves that when added together give the complex wave.   If we know the frequencies of each of those sine waves we know enough to rebuild our original signal. In the more practical sense if we have as many frequencies as we originally had samples we have a very close approximation of our original signal expressed in a different from. The process of turning samples into frequencies is called a Fourier Transform. Time Domain → Frequency Domain A piano as a frequency analyzer. |
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Fourier Transform
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De
Moivre's Theorem What is it What does it do?       The mathematically inclined should check out MATHEMATICS OF THE DISCRETE FOURIER TRANSFORM (DFT) at Stanford's Center for Computer Research in Music and Acoustics (CCRMA) |
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z and the Z-Transform
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z = eiω Z Transform |