Summary of lecture 1. E x = E x =T. X T (e i!t ) which motivates us to define the energy spectrum Φ xx (!) = jx (i!)j 2 Z 1 Z =T. 2 d!

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1 Summary of lecture I Continuous time: FS X FS [n] for periodic signals, FT X (i!) for non-periodic signals. II Discrete time: DTFT X T (e i!t ) III Poisson s summation formula: describes the relationship between FT X (i!) and DTFT X T (e i!t ) IV Sampling theorem: If there is no frequency content higher than! s = then X (i!) = X T (e i!t ) for! [! s = ;! s =]. V Aliasing: When frequency content >! s = appears under false name. VI From Parseval s formula we have E x = E x =T Z jx(t)j dt = X k= jx[k]j = Z jx (i!)j d! Z =T =T X T (e i!t ) d! which motivates us to define the energy spectrum Φ xx (!) = jx (i!)j Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 / 7

2 Outline Lecture Frequency description continued I Truncation! truncated DTFT II Leakage III Window functions IV Discrete frequency variable! Discrete Fourier Transform (DFT) V : FT! : DTFT! 3: Truncated DTFT! 4: DFT VI Circular convolution VII Zero-padding Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 / 7

3 Example: Rectangular window. x(t) x[k].8.6 X T(e iωt ) Time r[k] R T(e iωt ) Time Time xn[k] = x[k]r[k] X (N) T (e iωt ) Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 3 / 7

4 Example: Frequency resolution/separation DTFT for rectangular window of length N 8 7 N = N = N = N = Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 4 / 7

5 Example: Sum of two sines, rectangular window X (N) T (ei!t ) when x(t) = sin(:t) + sin(:9t) 8 7 N = 8 4 N = N = N = Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 / 7

6 Example: Window functions Comparison of time windows, and their respective DTFT:s Rect Triang Time Hann Hamm Time 3 Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 6 / 7

7 Example: Window functions x[k] = sin (:k) + : sin (:6k), T =, N = Rect Triang Hann Hamm.. Freq [Hz].. Freq [Hz].. Freq [Hz].. Freq [Hz] With a rectangular window it is difficult to see the sine with lower amplitude (i.e. lower energy). Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 7 / 7

8 DFT in Matlab In Matlab there is an efficient implementation of DFT called fft. X = fft(x); omega = (*pi/n/t)*(:n-); plot(omega,abs(x)) This plots the DFT for discrete frequencies in the interval [! s ] rad/s. The DFT is symmetric, i.e. X [N n] = X [n]. We can plot the DFT for discrete frequencies in the interval [! s =! s =] rad/s as follows. X = fftshift(fft(x)); omega = (*pi/n/t)*(-n/:n/-); plot(omega,abs(x)) We can also plot the frequency in Hz, f=omega//pi. Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 8 / 7

9 DFT in DSP Straightforward implementation using matrix multiplication function [X,f]=dft(x,T,f); N=length(x); if nargin<; T=; end if nargin<3; f=[:n-] /N/T; end W=exp(-i**pi*f(:)*[:N-]*T); X=W*x(:); end Complexity N (multiplications) Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 9 / 7

10 FFT in DSP Fast Fourier Transform: Fast implementation using recursive calls utilizing geometric properties of the exponential function. function X=fastdft(x) x=x(:); N=length(x); if log(n)~=round(log(n)), error( N must be a power of two ), end if N==; X=[x()+x();x()-x()]; else Xe=fastdft(x(::end)); Xo=fastdft(x(::end)); X=[Xe;Xe]+exp(-i**pi/N*(:N-) ).*[Xo;Xo]; end Complexity N log (N) (multiplications) Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 / 7

11 DTFT in DSP Approximation of the DTFT using zero-padding and FFT function [X,f]=dtft(x,T,N) if nargin<, T=; end if nargin<3, N=4; end X=fft([x(:); zeros(n-length(x),)]); f=(:n-) /N/T; Complexity for N = 4 is 4 (multiplications) Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 / 7

12 : FT! : DTFT! 3: Truncated DTFT! 4: DFT Example: x(t) is a sum of five cosine-functions. : X(iω) ωs : X T(e iωt ).8.8 Aliasing Frequency ω [rad/s] 3 : X (N) T (e iωt ) Frequency ω [rad/s] 4 : X[n] Leakage... 3 Frequency ω [rad/s]... 3 Frequency π NT n [rad/s] Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 / 7

13 Important properties of the FT and DTFT Important transform relationships (cf. Lecture ): x(t)y(t) $ X (i!)y (i!) $ Z Z X (i! i )Y (i )d x[k]y[k] $ T x(t )y()d T X T (e i!t )Y T Unfortunately these do not hold for the DFT! Z T X T e i(! )T Y T it e d T e i!t $ X x[k m]y[m] m= This is problematic because the DFT is what we can use in practice. However, we can understand the problem, and therefore handle it. Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 3 / 7

14 Linear and circular convolution Linear convolution X z[k] = x[k] y[k] = `= x[`]y[k `] k = ; : : : ; For finite sequences of length N x and N y : Assume x[k] and y[k] are zero outside k = ; : : : ; N x and k = ; : : : ; N y, respectively. z[k] = x[k] y[k] = NX x `= Compare to circular convolution (N x = N y = N) z[k] =x[k] y[k] = N X `= = IDFT (DFT (x[k]) DFT (y[k])) x[`]y[k `] k = ; : : : ; N x + N y x[`]y[(k `) mod N ] k = ; : : : ; N Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 4 / 7

15 Example: linear and circular convolution.9.8 x[k]..4 y[k] k x[k] y[k] k x[k] y[k] k k Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 / 7

16 Zero-padding for convolution Zero-pad signals to length N! x[k] y[k] = x z [k] y z [k].9.8 x z[k]..4 y z[k] k k x[k] y[k] k k x z[k] y z[k] Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 6 / 7

17 Zero-padding increases frequency grid resolution x[k] = sin(:! k) + sin(:9! k), y[k] = sin(:! k), N = 3, T =, zero-pad to length M.! = NT M =N X[n] Y[n] M =N X[n] Y[n] Frequency ( n NT ) [Hz] M =4N X[n] Y[n] Frequency ( n NT ) [Hz] M =8N X[n] Y[n] Frequency ( n NT ) [Hz] Frequency ( n NT ) [Hz] After zero-padding we can see the difference between x[k] and y[k]. Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 7 / 7

18 Summary of Lecture Truncation: We only have a finite number of samples in practice, which leads us to the truncated DTFT. Leakage: Truncation results in energy content leaking to nearby frequencies. Manifests itself in smeared peaks and limited frequency separation. Frequency separation: The possibility to separate two adjacent frequencies. Approx 4=(NT ). Time windows: We see the transform through a window. Several different windows available. DFT: the most practically useful Fourier transform. Circular convolution: For the DFT we have circular convolution and not standard linear convolution. Zero-padding: Add zeros to signal before computing DFT. Makes circular convolution coincide with linear convolution, and increases the frequency grid resolution. Fredrik Gustafsson (LiU) Digital Signal Processing, Lecture 7 8 / 7