Uniqueness of the Fourier Transform

The best way to prove the uniqueness of the Fourier transform is to exploit the orthogonality property of sinusoids, combined with some basic linear algebra. We can express the DFT as a matrix multiplication  

Usually, h(n) is an impulse response of a linear time-invariant system, and x(n) is the input signal to that (see Chapter 3). Also, h(n) and x n) are usually assumed to start at time = 0 that is, their value is zero for negative values of n. This property is called causality. So for causal signals x and h, Equation A.5 can be rewritten as  

If x n) has finite length P, and h n) has finite length M, then we can fnrther limit the convolution summation to  

This is because computing any points beyond those boundaries would result in a zero result. Note that jc and h can be swapped and the equation result is the same (this is called commuting, and the property is called commutativity). This means that it makes no difference which signal corresponds to the system or which is the input. Thus, singing in a room (convolution of a voice with the room impulse response) is mathanatically identical to a room impulse response singing through the voice. 

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