Digital Signal Processing, Lecture 2 Frequency description continued, DFT

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1 Outline cture 2 2 Digital Signal Processing, cture 2 Frequency description continued, DFT Thomas Schön Division of Automatic Control Department of Electrical Engineering Linköping i University it schon@isy.liu.se Phone: Office: House B, entrance Summary of l 2. Finish sampling in the time domain (approximation 1-2) 3. Finite (truncated) signals 4. Windows and leakage (approximation 2-3) 5. DFT a) Circular convolution b) Zero-padding 6. Truncated DTFT DFT (approximation 3-4) Summary of cture 1 (I/III) 3 Summary of cture 1 (II/III) 4 We saw a few examples of how signal processing can be used both in itself and within a bigger context. Using Parseval s formula we have Transform Overview Periodic signals Continuous-time Fourier series Discrete-time DFT (more about this today) Non-periodic signals Continuous-time Fourier transform (FT) Discrete-time Discrete time Fourier transform (DTFT) The DTFT looks like the FT, but it is periodic. Signal Periodic L1-signal General Continuous time FS FT Laplace Discrete time DFT DTFT ZT which motivates the following definition of the energy spectrum Approximations FT DTFT Truncated DTFT DFT

2 Summary of cture 1 (III/III) 5 Motivating Example Frequency Domain 6 Approximation 1-2 Poisson s summation formula states: What are the implications of this? Time (s) Solid line measured current in a 400 kv transformer (Söderåsen) when it is switched on. Dashed line Approximation based on the fundamental frequency and the 3 first overtones (harmonics). Time (s) Showing the fundamental frequency (50 Hz) and the three first overtones. Hence, a compact and simple representation in the frequency domain. Fourier Series Example (Square Wave) 7 Aliasing in the UAV Application 8 Using more and more sinusoidals gives a better approximation as expected. Investigating the measurements from the gyroscopes (angular velocities) Hz = 830 rpm (rotor speed) Aliasing from Hz = 6384 rpm (Engine rpm) Figure courtesy of the Unmanned Aircraft Systems Technologies Lab, Department of Computer and Information Science, Linköping University Hz = 1660 rpm (double rotor speed) Nyquist frequency (half the sampling frequency): 100 Hz

3 Aliasing for the Chirp Signal 9 Problems with Truncation akage (I/II) 10 The chirp signal is defined as N = 15000; y = sin(2*pi*(0:n-1).^2./n); sound(y); Spectogram t us study a sinusodial with frequency 1 rad/s MATLAB k = 0:1:15; x = sin(k); plot(k,x); The instantaneous frequency is We expect a straight line here Problems with Truncation akage (II/II) 11 Problems with Truncation Limited Separation 12 The approximated DTFT (i.e., the truncated t DTFT) The truncated t DTFT for the signal is shown below for three different N Main loob MATLAB [X,w] = dtft(x); plot(w,abs(x)); (X)) MATLAB [X,w] = dtft(x); plot(w,abs(x)); Side loobs Note that we plot the absolute value of the DTFT The sin is leaking to neighbouring frequencies! The leakage results in limited frequency separation.

4 Window Functions 13 Discrete Fourier Transform (DFT) 14 From a practical point of view the DFT is the most useful Fourier transform We see the transform through the frequency window Definition (DFT) Time domain Frequency domain Rectangular window Triangular window Rectangular window Triangular window The signal x[k] and its transform X[n] are defined at the following times and frequencies: In MATLAB where the frequency grid resolution is X = fft(x); omega = 2*pi*(0:N-1) /T/N; plot(omega,abs(x)) Basic Time Frequency Relations for the FT 15 Circular Convolution (I/III) 16 Some of the most important t symmetry (duality) relations between the time and the frequency domain for the FT are Convolve the sequences x (approximation of ideal LP-filter) and y (ramp) showed below: These are not all valid for the DTF. This is a problem, since we always are working with finite data in practice. However, if we understand the problem, we can handle it.

5 Circular Convolution (II/III) 17 Circular Convolution MATLAB Code (III/III) 18 Linear convolution Circular convolution k=-31:32; x=sin(.5*k)./k; x(32)=0.5; plot(x), hold on, stem(x), hold off title('x') y=1:64; y=y/64; plot(y), hold on, stem(y), hold off title('y') z=conv(x,y); plot(z), hold on, stem(z), hold off title('z=x*y') zc=real(ifft(fft(y).*fft(x))); plot(zc), hold on, stem(zc), hold off title('zc=ifft(fft(y).*fft(x)) ) Denser Frequency Grid by Zero-Padding 19 User Aspects 20 It is important that The user understand the fundamental limitations of the DFT, which can be split up into three possible problems: Operation Transform Limitation Problem Time sampling DTFT Alias Truncation DTFT Frequency separation akage Frequency sampling DFT Circular convolution N = 32; T = 1; omega0 = 2*pi/(N*T); k = 0:1:N-1; x1 = sin(5.1*omega0*k) + sin(5.9*omega0*k); subplot(3,1,1) plot(abs(fft(x1))) hold on; stem(abs(fft(x1))); hold off; subplot(3,1,2) plot(abs(fft([x1 zeros(1,32)]))) hold on; stem(abs(fft([x1 zeros(1,32)]))); hold off; subplot(3,1,3) plot(abs(fft([x1 zeros(1,96)]))) hold on; stem(abs(fft([x1 zeros(1,96)]))); hold off; Zero-padding the signal before computing the DFT first increases the grid density of the frequency axis and also makes circular convolution coincide with linear convolution. Choice of coordinate axis (linear/logaritmic, Hz/(rad/s), the DFT is symmetric, X[n] = X[N-n]), see the book for details and the exercises for practice.

6 A Few Concepts to Summarize cture 2 Truncation: In practice we only have finitely many samples, implying ing that signals are truncated, motivating the truncated DTFT. akage: Truncation implies that the energy content leaks to neighbouring frequencies, which manifests itself in smeared peaks and limited frequency separation. Frequency separation: The possibility to separate two adjacent frequencies, roughly. Windows: We see the transform through a window. There are several window functions to choose from. Circular convolution: The standard linear convolution is not valid for the DFT, instead we had to define circular convolution. Zero-padding: Amounts to adding zeros to the signal before applying the DFT. This makes the circular convolution coincide with the linear convolution. It also first increases the grid density of the frequency axis. 21