The spectrum of a signal is given by its Fourier transform. In order to compute it, we need all the samples of the signal. Therefore, the spectrum is a characteristic of the
whole signal. However, in many applications (e.g., in speech processing) one needs to find the spectrum of a short section of the signal. In order to do this, we define a length-N sliding window x_i(n) of the signal x(n) as

That is, we take N samples of x(n) starting from position i backwards. We then define the short-time spectrum of x(n) at the position i as the DFT of x_i(n); that is:

(c) Discuss whether it is advantageous or not, in terms of arithmetic operations, to use the above formula for H_k (z) to compute a linear convolution using the overlap-and-add method. Repeat this item for the overlap-and-save method. A good coverage of recursive computations of sinusoidal transforms can be found in Liu et al. (1994).

Transcribed Image Text:

xi(n) = x(n+i-N+1), for