Applications of digital signal processing - Continuous and Discrete Time Signals and Systems - page 748

Note that the phase of x 2 [ k ] is given by θ 0

= 0 . 0001 π k 2 . By differentiating
the phase θ 0 with respect to k , the instantaneous frequency is obtained as
ω 0 = 0 . 0002 π k . The instantaneous frequency ω 0 is a function of time k ,

and increases proportionately as k increases. However, this time-varying
nature of the frequency is not obvious from the magnitude spectrum shown
in Fig. 17.1(c). Since the DFT averages the frequency components over all
time k , the DFT provides a misleading result in this case.

Example 17.1 shows that the DFT magnitude spectrum based approach is
convenient for estimating the spectral content of a stationary signal comprising
sinusoidal components with fixed frequencies. However, it may provide mis-
leading results for non-stationary signals, where the instantaneous frequency
changes with time. In other words, it is difficult to visualize the time evolution
of frequency in the DFT magnitude spectrum. The short-time Fourier transform
is defined in Section 17.1.1 to address this limitation of DFT.

17.1.1 Short-time Fourier transform

In order to estimate the time evolution of the frequency components present in
a signal, the short-time Fourier transform (STFT) parses the signal into smaller
segments. The DFT of each segment is calculated separately and plotted as a
function of time k . The STFT is therefore a function of both frequency Ω and
time k . Mathematically, the STFT of a DT signal x [ k ] is defined as follows:
X s ( Ω , b ) = ∞ x [ k ] g ∗ [ k − b ]e − j Ω k ,

(17.1)

k =−∞

where the subscript s in X s ( Ω , b ) denotes the STFT and b indicates the amount
of shift in the time-localized window g [ k ] along the time axis. Typical windows
used to calculate the STFT are rectangular, Hanning, Hamming, Blackman, and
Kaiser windows. Compared to the rectangular window, the tapered windows,
such as Hanning and Blackman, reduce the amount of ripple and are generally
preferred.
In most cases, the time shift b is selected such that successive STFTs are taken
over adjacent samples of x [ k ] and there is some overlap of samples between
successive STFTs. As discussed earlier, the STFT is a function of two variables:
the frequency Ω and the central location of the window. It is typically plotted
as an image plot, known as a spectrogram, with frequency Ω varying along
the y -axis and the time (i.e. the center of the window function) varying along
the x -axis. The intensity values of the image plot show the relative strength of
various frequency components in the original signal.

Example 17.2
Plot the spectrogram of the signal

x 2 [ k ] = cos(0 . 0001 π k 2 ) for duration

k = [0 , 39 999].