1. 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

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

3. … 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.

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

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

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

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

8. 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

9. 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)

10. Example Consider the signal

11. … zoom in Consider the signal

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

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

14. … zoom in Consider the signal

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

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

17. 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!!!

18. 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

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

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