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Where mk is the integer index of the largest magnitude sample |X(mk)|. Where real(d) means the real part of the d correction factor defined as The signal's index-based center frequency, mpeak, can be estimated using FFT spectral magnitudes of a narrowband signal. The vertical magnitude axis is linear, not logarithmic.įigure 13-38. Here we describe a computationally simple frequency estimation scheme.Īssume we have FFT spectral samples X(m), of a real-valued narrowband time signal, whose magnitudes are shown in Figure 13-38. Many other techniques for enhanced-precision frequency measurement have been described in the scientific literature-from the close-to-home field of geophysics to the lofty studies of astrophysics-but most of those schemes seek precision without regard to computational simplicity. With the spectral peak located at bin mpeak = 21, we estimate the signal's center frequency, in Hz, usingīoth schemes, collect more data and zero-padding, are computationally expensive. That would also provide an improved frequency resolution of fs/4N, as shown in Figure 13-37(b). Or we could pad (append to the end of the original time samples) the original N time samples with 3N zero-valued samples and perform a 4N-point FFT on the lengthened time sequence. We could collect, say, 4N time-domain signal samples and perform a 4N-point FFT yielding a reduced bin spacing of fs/4N. Spectral magnitudes: (a) N-point FFT (b) 4N-point FFT. We often need better frequency estimation resolution, and there are indeed several ways to improve that resolution.įigure 13-37. In this situation, our frequency estimation resolution is half the FFT bin spacing. The real-valued sinusoidal time signal has, in this example, a frequency of 5.25fs/N Hz. The FFT bin spacing is fs/N where, as always, fs is the sample rate.) Close examination of Figure 13-37(a) allows us to say the sinusoid lies in the range of m = 5 and m = 5.5, because we see that the maximum spectral sample is closer to the m = 5 bin center than the m = 6 bin center. (Variable m is an N-point FFT's frequency-domain index. There we see the sinusoid's spectral peak residing between the FFT's m = 5 and m = 6 bin centers. As such, due to the FFT's leakage properties, the discrete spectrum of a sinusoid having N time-domain samples may look like the magnitude samples shown in Figure 13-37(a). Upon applying the radix-2 fast Fourier transform (FFT), our narrowband signals of interest rarely reside exactly on an FFT bin center whose frequency is exactly known. In the practical world of discrete spectrum analysis, we often want to estimate the frequency of a sinusoid (or the center frequency of a very narrowband signal of interest).