We know how to acquire analog signals for digital processing (pre-filtering, sampling, and A/D conversion) and to compute spectra of discrete-time signals (using the FFT algorithm), let's put these various components together to learn how the spectrogram shown in Figure 1, which is used to analyze speech, is calculated. The speech was sampled at a rate of 11.025 kHz and passed through a 16-bit A/D converter.

The resulting discrete-time signal, shown in the bottom of Figure 1, clearly changes its character with time. To display these spectral changes, the long signal was sectioned into frames: comparatively short, contiguous groups of samples. Conceptually, a Fourier transform of each frame is calculated using the FFT. Each frame is not so long that significant signal variations are retained within a frame, but not so short that we lose the signal's spectral character. Roughly speaking, the speech signal's spectrum is evaluated over successive time segments and stacked side by side so that the xx-axis corresponds to time and the yy-axis frequency, with color indicating the spectral amplitude.

An important detail emerges when we examine each framed signal (Figure 2). At the frame's edges, the signal may change very abruptly, a feature not present in the original signal. A transform of such a segment reveals a curious oscillation in the spectrum, an artifact directly related to this sharp amplitude change. A better way to frame signals for spectrograms is to apply a window: Shape the signal values within a frame so that the signal decays gracefully as it nears the edges. This shaping is accomplished by multiplying the framed signal by the sequence wn w n . In sectioning the signal, we essentially applied a rectangular window: wn=1 w n 1 , 0≤n≤N-1 0 n N1 . A much more graceful window is the Hanning window; it has the cosine shape wn=121-cos2πnN w n 1 2 1 2 n N . As shown in Figure 2, this shaping greatly reduces spurious oscillations in each frame's spectrum. Considering the spectrum of the Hanning windowed frame, we find that the oscillations resulting from applying the rectangular window obscured a formant (the one located at a little more than half the Nyquist frequency).

If you examine the windowed signal sections in sequence to examine windowing's affect on signal amplitude, we see that we have managed to amplitude-modulate the signal with the periodically repeated window (Figure 3). To alleviate this problem, frames are overlapped (typically by half a frame duration). This solution requires more Fourier transform calculations than needed by rectangular windowing, but the spectra are much better behaved and spectral changes are much better captured.

The speech signal, such as shown in the speech spectrogram, is sectioned into overlapping, equal-length frames, with a Hanning window applied to each frame. The spectra of each of these is calculated, and displayed in spectrograms with frequency extending vertically, window time location running horizontally, and spectral magnitude color-coded. Figure 4 illustrates these computations.