Fourier Analysis and the Speech Spectrogram
Background Information
The Fourier Transform is often introduced to students as a construct to evaluate both continuous- and discrete-time signals in the frequency domain. It is first shown that periodic signals can be expressed as sums of harmonically-related complex exponentials of different frequencies. Then, the Fourier Series representation of a signal is developed to determine the magnitude of each frequency component's contribution to the original signal. Finally, the Fourier Transform is calculated to express these coefficients as a function of frequency. For the discrete-time case, the analysis equation is expressed as follows:
$ X(e^{j\omega}) = \sum_{n=-\infty}^\infty x[n] e^{-j\omega n} $
This expression yields what is commonly referred to as the "spectrum" of the original discrete-time signal, x[n]. To demonstrate why this is the case, consider the following discrete-time function:
$ x[n] = cos(2 \pi 10 t ) $
Applying the analysis equation above yields the following Fourier Transform of the signal:
$ X(e^{j\omega}) = \pi \delta (\omega - 20\pi) + \pi \delta (\omega + 20\pi) $
Intuitively, this expression shows that a cosine function is the sum of two complex exponentials with fundamental frequencies of -10 and +10 Hertz. Indeed, Euler's Formula provides a way to rewrite the cosine function in this form.
Applying the Fourier Transform
