1. Real time DSP Professors: Eng. Julian S. Bruno Eng. Jerónimo F. Atencio Sr. Lucio Martinez Garbino

2. Discrete Time Fourier Transform DTFT N

3. Discrete Fourier Series (I)

4. Discrete Fourier Series (II)

5. Discrete Fourier Series (III) DFS coefficients DTFT of periodical signal

6. Discrete Fourier Transform (I) Sampling ωk=2πk/N x[m] is an aperiodic sequence DFS

7. Discrete Fourier Transform (II)

8. Discrete Fourier Transform (III) 0 ≤ k ≤ N-1 0 ≤ n ≤ N-1 The inherent periodicity is always present Propierties: Circular Shift of a Sequence Circular Convolution

9. Sampling the Fourier Transform DFT Sampling DTFT DFS

10. DFT Propierties The circular convolution corresponding to X 1 [k]X 2 [k] is identical to the linear convolution corresponding to X 1 (e jw )X 2 (e jw ) if N, the length of the DFTs, satisfies N ≥ L + P - 1. Circular Shift of a Sequence Circular Convolution

11. Implementing Linear Time-Invariant Systems Using the DFT y[n] = x[n] * h[n] h [n] x [n]

12. Overlap-add method y r [n] = x r [n] * h[n]

13. Overlap-save method y r [n] = x r [n] * h[n]

14. Understanding the DFT Equation In this example we have a 4 samples signal and we use DFT to get its frequency representation. The result for each frequency component is obtained after computing 8 real sums and multiplications.

15. DFT example (I) Consider a signal formed with 2 sinusoidal, one of 1 KHz and the other of 2 KHz and a phase shift of ¾π. N = 8 samples. Fs = 8000 samples/s. Fs/N = 1Khz First computations are showed in detail.

16. DFT example (II)

17. DFT example (III)

18. DFT example (IV) Here we show the final result in both representations formats. The complex DFT outputs for m=1 to m=(N/2)-1 are redundant with frequency output values form m>(N/2) We can see an even symmetry in Magnitude and Real representations, while an odd symmetry in Imaginary and Phase. It can be verified the amplitude and phase relationship between the sinusoidal components, but absolute values? Fixed point DSP

19. DFT Leakage (I) If there are frequency components that are not integer multiples of f res, we got leakage. Leakage evidences the effect of sampling during finite (and rectangular) time window. Leakage is an unavoidable fact of life when we perform the DFT on real world finite-length time sequences

20. DFT Leakage (II) As can be seen, the sinc function is always present, but only evidenced when frequency components are not integer multiples of f res. The DFT output is a sampled version of the continuous spectral

21. Time Windowing (I) The only fact of considering a finite length time sequence, is equivalent to convolve a sinc with all frequency samples. If we use a window, we will convolve with the spectrum this window. The net effect of windowing is a better spectral estimation, reducing leakage and picket fence effect.

22. Time Windowing (II) Spectral analysis: Equivalent Noise Bandwith Processing Gain Overlap Correlation Scalloping Loos Worst Case Processing Loss Minimun Resolution Bandwidth

23. Equivalent Noise Bandwith

24. Processing Gain

25. Overlap Correlation

26. Picket fence effect The picket fence effect is a manifestation of applying DFT over a finite time sequence. The net effect of windowing is a smoothed frequency response of a sinc at each frequency index.

27. Minimun Resolution Bandwidth

28. Time Windowing (II) The window selection is a trade-off between main lobe widening, first sidelobe levels, and how fast the sidelobes decrease with increased frequency.

29. Goertzel Algorithm The Goertzel algorithm is a digital signal processing technique for identifying frequency components of a signal, published by Dr. Gerald Goertzel in 1958 MethodReal multiplies Real additions Single-bin DFT4N2N FFT2Nlog 2 NNlog 2 N GoertzelN+22N+1 If you implement the Goertzel algorithm L times to detect L different tones, Goertzel is more efficent than FFT when L< log 2 N

30. Goertzel Algorithm Implementation

31. Zoom FFT

32. Zero Stuffing Zero stuffing is a way of increasing frequency resolution. The spectrum visualized corresponds to the convolution of a sinusoidal and a rectangular signal. Thus, the underlying spectrum of the sinusoidal is distorted by a sinc.