Class 7: Examples and exercises

Transcription

1 et Digital signal processing Class 7: Eamples and eercises Rocio Arroyo Valles, Geert Leus, and Alle-Jan van der Veen Faculty of EEMCS, Delft University of Technology Delft University of Technology Challenge the future

2 Outline Eercise 1: generation and analysis of random noise signals Eercise 2: generation and analysis of harmonic signals Eercise 3: generation and analysis of nonstationary signals Break (15 min) Eercise 4: design and analysis of elementary digital filters Eercise 5: filtering and analysis of random noise signals Challenge: filter design for interference cancellation in speech 2

3 Eercise 1 Generation and analysis of random noise signals Background [Hayes, Sec ]: A white noise signal v(n) is characterized by an impulse autocorrelation function (ACF) and a flat power spectral density (PSD) 3

4 Eercise 1 Generation and analysis of random noise signals MATLAB Eercise: 1. Generate and plot 1000 samples of zero mean unit variance white noise 2. Calculate and plot the sample ACF for lag < Calculate and plot the PSD 4. Split the white noise signal into 10 segments of 100 samples each 5. Calculate and plot the average sample ACF ( lag <100) 6. Calculate and plot the average PSD 4

5 Eercise 2 Generation and analysis of harmonic signals Background [Hayes, Eamples and 3.3.3]: A sum of sinusoids with uniformly distributed random phases has an ACF that is also a sum of sinusoids and a PSD consisting of a sum of impulse functions 5

6 Eercise 2 Generation and analysis of harmonic signals Background [Hayes, Sec ]: The autocorrelation matri of a wide-sense stationary random process is a Hermitian Toeplitz matri containing the different ACF values MATLAB Eercise: 1. Generate and plot 2048 samples of a sum of M=10 sinusoids with unit amplitudes, uniformly distributed random phases, and frequencies m = m*(2*pi)/64 2. Calculate and plot the sample ACF for lag < Calculate and plot the PSD 4. Construct the autocorrelation matri and plot its eigenvalues 6

7 Eercise 3 Generation and analysis of nonstationary signals Background: Most of the theory on random processes is based on the assumption of wide-sense stationary signals 1. mean does not depend on time: 2. ACF does not depend on time, only on lag: 3. variance is finite: However, many physical signals are nonstationary by nature. The short-time Fourier transform (STFT) is a mathematical tool to estimate the time-varying spectrum of nonstationary signals. The STFT is obtained by splitting a signal into shorter overlapping segments, and calculating a discrete Fourier transform (DFT) for each segment. The STFT can be visualized in a spectrogram. 7

8 Eercise 3 Generation and analysis of nonstationary signals MATLAB Eercise: 1. ad the sound file speech_dft.wav (included in Simulink DSP Blockset) into a vector in the MATLAB workspace, and determine the sampling rate 2. Plot and play back the time-domain signal 3. Plot the spectrogram, using the following parameters: length of segments = 256 samples overlap of segments = 128 samples length of segment DFT = 256 samples visible frequency range = half sampling rate time on -ais, frequency on y-ais 8

9 Eercise 4 Design and analysis of elementary digital filters Background [Hayes, Sec. 3.6]: First-and second-order all-zero filters: H(z) = b 0 + b 1 z -1 + b 2 z -2 LOWPASS Im b 1 /b 0 >0 b 2 =0 BANDSTOP Im b 12 <4b 0 b 2 b 2 0 HIGHPASS Im b 1 /b 0 <0 b 2 =0 9

10 Eercise 4 Design and analysis of elementary digital filters Background [Hayes, Sec. 3.6]: First-and second-order all-pole filters: H(z) = b 0 /(1 + a 1 z -1 + a 2 z -2 ) LOWPASS Im a 1 <0 a 2 =0 BANDPASS Im a 12 <4a 2 a 2 0 HIGHPASS Im a 1 >0 a 2 =0 10

11 Eercise 4 Design and analysis of elementary digital filters Background [Hayes, Sec. 3.6]: Biquadratic filters: H(z) = (b 0 + b 1 z -1 + b 2 z -2 )/(1 + a 1 z -1 + a 2 z -2 ) Special case: b 0 = 1 b 1 = -2r z cos( c ) b 2 = r 2 z a 1 = -2r p cos( c ) a 2 = r 2 p BANDPASS Im r z <r p BANDSTOP Im r z >r p 11

12 Eercise 4 Design and analysis of elementary digital filters MATLAB Eercise: 1. Design a highpass filter with one zero at z= Plot the pole-zero diagram in the comple plane 3. Plot the frequency response (magnitude & phase) 4. Design a bandpass filter with central frequency c =1 rad, r z = 0.8, r p = Plot the pole-zero diagram in the comple plane 6. Plot the frequency response (magnitude & phase) 12

13 Eercise 5 Filtering and analysis of random noise signals Background [Hayes, Sec. 3.4]: Filtering a signal (n) in a filter with impulse response h(n) yields In the frequency domain, this is equivalent to 13

14 Eercise 5 Filtering and analysis of random noise signals MATLAB Eercise: 1. Generate and plot 1000 samples of zero mean unit variance white noise 2. Calculate and plot the PSD 3. Design a highpass filter with one zero at z= Plot the frequency magnitude response 5. Filter the white noise signal using the highpass filter 6. Calculate and plot the PSD of the resulting output signal 7. Design a bandpass filter with central frequency c =1 rad, r z = 0.1, r p = Plot the frequency magnitude response 9. Filter the white noise signal using the bandpass filter 10. Calculate and plot the PSD of the resulting output signal 14

15 Challenge Filter design for interference cancellation in speech MATLAB Eercise: 1. ad the sound file speech_dft.wav (included in Simulink DSP Blockset) into a vector in the MATLAB workspace 2. Filter the sound signal with the filter H(z) = 1/( z z z z z z -7 ) This is an unstable filter which causes a sinusoidal oscillation in the speech signal (acoustic feedback) 3. Challenge: analyze the speech signal corrupted with acoustic feedback, and design an appropriate bandstop filter that is capable of cancelling the sinusoidal oscillation while affecting the speech signal as little as possible 15