Since all real signals are continuous, Gibbs phenomenon does not occur, and we can thus assume the Fourier series representation is identical to the periodic signal. The peak is independent of N, though the period of the oscillatory overshoot is a function of N. However, on each side of the discontinuity, the Fourier series has oscillatory overshoot with a period of T 0/2 N, where T 0 is the signal period and N represents the number of terms included in the Fourier series, and a peak value of almost 9% of the amplitude of the discontinuity. In that, the Fourier series at the point of discontinuity converges to an average of the left-hand and right-hand limits of g( t) at the instant of discontinuity (i.e., to the arithmetic mean of the signal value on either side of the discontinuity). When there is a discontinuity in a piecewise continuously differentiable periodic signal g( t), the series exhibits a behavior known as Gibbs phenomenon, as shown in Figure 3.22.
Some of the best results in audio compression are based on sophisticated generalizations of this spectrogram technique-referred to either as lapped transforms or as local cosine expansions, see Malvar 1992 and Mallat 1998.
For example, by discarding all of the spectrogram values having magnitudes less than 1/320 times the largest magnitude spectrogram value, the young girl's version of “twinkle, twinkle, little star” can be approximated, without noticeable degradation of quality, using about one-eighth the amount of data as the original recording. After discarding (setting to zero) all the values in the spectrogram with magnitudes below a threshold value, the inverse procedure creates an approximation to the signal which uses significantly less data than the original signal. One application of this inverse procedure is to the compression of audio signals. In other words, we can recover the original signal by inverting the succession of DFTs that make up its spectrogram. (b) Similar graphs for the author's rendition of “twinkle, twinkle, little star.” (a) Bottom graph displays data from a recording of a young girl singing “twinkle, twinkle, little star.” Top graph displays the spectrogram magnitudes for this recording.
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