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Published byLouisa Burke Modified over 2 years ago

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Let’s go back to this problem: We take N samples of a sinusoid (or a complex exponential) and we want to estimate its amplitude and frequency by the FFT. What do we get? Estimate the Frequency Spectrum

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Take the FFT … FFT Best Estimates based on FFT: Frequency: Amplitude: How good is this estimate?

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… again recall what we did… Take a complex exponential of finite length: then its DFT looks like this where we define This is important to understand how good the spectral estimate is.

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See the plot of W N / N Main Lobe Side Lobes

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See the plot of W N / N in dB’s Main Lobe Side Lobes dB

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… and zoom around the main lobe N=64N=256N=1024

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Main Lobe The width of the Main Lobe decreases as the data length N increases

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Side Lobes Sidelobes are artifacts which don’t belong to the signal. As the data length N increases, the height of the sidelobes stays the same; the height of the first sidelobe is 13dB’s below the maximum

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Effect on Frequency Resolution Why all this is important? 1. It has an effect on the frequency resolution. Suppose you have a signal with two frequencies and you take the DFT. See the mainlobes: you can resolve them (2 peaks) you cannot resolve them (1 peak)

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Example Consider the signal

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… zoom in Consider the signal

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Another Example Consider the signal Only One Peak: Cannot Resolve the two frequencies!!!

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… take more data points … … of the same signal Two Peaks: Can Resolve the two frequencies.

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… zoom in Consider the signal

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Now the Sidelobes Consider the signal These are all sidelobes!!!

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… add a low power component Consider the signal Because of sidelobes, cannot see the low power frequency component.

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Why we have sidelobes? There reason why there are high frequency artifacts (ie sidelobes) is because there is a sharp transition at the edges of the time interval. Remember that the signal is just one period of a periodic signal: One Period Discontinuity!!!

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Remedy: use a “window” A remedy is to smooth a signal to “zero” at the edges by multiplying with a window data hamming window windowed data

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Use Hamming Window Take the FFT of the “windowed data”: dB k

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Use Hamming Window … zoom in 1217 dB k Estimate two frequencies

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