<< Chapter < Page | Chapter >> Page > |
The DSTFT basis functions are of the form
where is a window function [link] . If these functions form an orthogonal (orthonormal) basis, . The DSTFT coefficients, , estimate the presence of signal components centered at in the time-frequency plane, i.e., the DSTFT gives a uniform tiling of the time-frequency plane with the basis functions . If and are time and frequency resolutions respectively of , then the uncertainty principle demands that [link] , [link] . Moreover, if the basis is orthonormal, the Balian-Low theorem implies either or is infinite. Both and can be controlled by the choice of , but for any particular choice, there will be signals for which either the time or frequency resolution is not adequate. [link] shows the time-frequency tiles associated with the STFT basis for a narrow and widewindow, illustrating the inherent time-frequency trade-offs associated with this basis. Notice that the tiling schematic holds forseveral choices of windows (i.e., each figure represents all DSTFT bases with the particular time-frequency resolutioncharacteristic).
The discrete wavelet transform (DWT) is another signal-independent tiling of the time-frequency plane suited for signals where high frequency signal componentshave shorter duration than low frequency signal components. Time-frequency atoms for the DWT, , are obtained by translates and scales of the wavelet function . One shrinks/stretches the wavelet to capture high-/low-frequency componentsof the signal. If these atoms form an orthonormal basis, then . The DWT coefficients, , are a measure of the energy of the signal components located at in the time-frequency plane, giving yet another tiling of the time-frequency plane. As discussed in Chapter: Filter Banks and the Discrete Wavelet Transform and Chapter: Filter Banks and Transmultiplexers , the DWT (for compactly supported wavelets) can be efficiently computedusing two-channel unitary FIR filter banks [link] . [link] shows the corresponding tiling description which illustrates time-frequency resolution propertiesof a DWT basis. If you look along the frequency (or scale) axis at someparticular time (translation), you can imagine seeing the frequency response of the filter bank as shown in [link] with the logarithmic bandwidth of each channel. Indeed, each horizontal strip inthe tiling of [link] corresponds to each channel, which in turn corresponds to a scale . The location of the tiles corresponding to each coefficient is shown in [link] . If at a particular scale, you imagine the translations along the axis, you see the construction of the components of a signal at that scale. Thismakes it obvious that at lower resolutions (smaller ) the translations are large and at higher resolutions the translations are small.
Notification Switch
Would you like to follow the 'Wavelets and wavelet transforms' conversation and receive update notifications?