Figure 5: Coherent power gain вЂ“ illustrated for various window functions Scalloping lossДЄs discussed above, the FFT (equivalently the DFT) operation can be seen as processing a signal through a set of filter banks with bandwidth О”f centered on the successive bin frequencies. Table 1 below, lists the coherent power gain for some well known window functions.

The length of the window is L = 151 points and the simulation assumes an oversampling factor of 16 (i.e, F s=160 Hz). In other words, the noise floor displayed at the FFT output, varies according to the window of choice.Д®quivalent noise bandwidth (ENBW), often specified in terms of FFT bins, is the bandwidth of a fictitious brick-wall filter such that the amount of noise power accumulated inside the brick-wall filter is same as the noise power accumulated when processing white noise through the selected window.ДЇigure 4: Windowing a 10 Hz sinusoidal signal with Hann window of length L = 151 (sampling frequency F s=160 Hz)Д¬oherent power gain of a window w of length L is given byДЇollowing plot depicts the coherent power gain (i.e, the reduction in the FFT magnitude when the input signal is processed with a window) of a windowed sinusoidal signal of frequency 10 Hz. In effect, the amplitude estimate in each frequency bin is influenced by the amount of accumulated noise contributed by the window function of choice. Hence the amount of noise that gets accumulated into each filter bank is affected by the bandwidth of the chosen window function. As a result, the choice of window function affects the amount of signal and noise that goes inside each filter bank. To suppress the spectral leakage, the signals are multiplied with a window function so as to smooth the discontinuity at the edges of the FFT slices. This phenomenon is called spectral leakage. This may lead to discontinuities at the edges of each slice, that causes the energy contained in each frequency bin to spill into other bins. When the time-limited signal slice is subjected to analysis using FFT algorithm (used for computing DFT), the FFT implicitly assumes that the signal essentially repeats itself after the observed interval. In reality, signals are of time-limited nature and nothing can be known about the signal beyond the measured interval. Figure 1: DFT (Discrete Fourier Transform) viewed as processing through a set of filter banks

Fft windowing