EE2S11 Signals and Systems, part 2 Ch.11 The Discrete-Time Fourier Transform (DTFT Contents definition of the DTFT relation to the -transform, region of convergence, stability frequency plots convolution property, filters inverse DTFT Skip sections 11.2.4.1 (decimation/interpolation, 11.3 (Fourier Series, 11.4 (DFT. These are covered in EE2S31. 1
i.e.,! Definition of the Discrete-time Fourier transform (DTFT " $ # ( Continuous function of (while is a time series : periodic in, period : It sufficies to consider the interval. is called the spectrum and measures the frequency content of., where : amplitude spectrum, : phase spectrum Sufficient condition for convergence of the infinite sum: ' &% is absolutely summable (. 2
( Relation of the -transform to the DTFT The relation to the -transform is obtained by setting (assuming that the unit circle is in de ROC. Hence, we often write as, cf. the book. (This notation avoids confusion between and, different functions. We immediately obtain (LTI systems: (filters! exists if the system is BIBO stable (, i.e., the unit circle is in the ROC of. (no ordinary DTFT because of ROC % 3
Frequency plots Plot the amplitude and phase spectrum of # (assume R Amplitude spectrum Phase spectrum ( -plane The response can also be estimated using phasors. 4
or 1 x[n] = (.5 n u[n] 1 x[n] = (.5 n u[n].5 2 4 6 8 1 n X(ω 2 1 π π/2 π/2 π ω 1 2 4 6 8 1 n X(ω 2 1 π π/2 π/2 π ω π φ(ω π φ(ω π π π/2 π/2 π ω π π π/2 π/2 π ω In matlab: use, use to resolve phase jumps of. 5
# Estimating frequency plots using phasors Given a transfer function, draw a plot of and. mod # To gain some insight: compute this for a number of values of. 4 2 a=.7 b= X(ω 2 1 a=.9 b=.8 X(ω π π/2 π/2 π ω φ(ω π π π π/2 π/2 π ω π π/2 π/2 π ω φ(ω π π π π/2 π/2 π ω 6
" Example: DTFT of a pulse!! " "! pulse of length! " -plane The amplitude spectrum is periodic sinc-function (Dirichlet-function with (L Hopital. The phase spectrum is # (lineair phase plus phase jumps of due to sign changes of. 7
" 1 5 X(ω ( π π/2 π/2 π ω φ(ω π π/2 π/2 π π π/2 π/2 π ω zero crossings for Phase slope Phase jumps ( delay change of sign The linear phase corresponds to a delay!, half the duration of the pulse. The first zero in the amplitude spectrum (right of the peak at gives an indication of the bandwidth (although this signal is not band limited: 8
Note: The duration of the pulse ( is inversely proportional to the bandwidth in the frequency domain (. If then, and (constant. If 'then (step and : the amplitude spectrum converges to an impulse. (Not allowed to take the limit because the phase does not converge see later for the DTFT of. 9
-transform ofthe Example: DTFT of a non-causal signal Determine the spectrum of the non-causal signal with %. Solution is (transform the causal/anti-causal parts separately: # with as ROC the intersection of the ROC of the causal and anti-causal part: ROC: % % The ROC contains the unit circle. Hence Note that is real-valued (. We have seen the plot of before... 1
Relation of the continuous-time Fourier Transform to the DTFT Consider a signal and sample it with period, The (continous-time Fourier transform is Set and. Then The definition of (spectrum of a time series is consistent to that of (spectrum of a continuous-time signal. 11
" with Inverse DTFT! The integral runs over 1 period of the spectrum. Proof: [ [ ] ] The same result is obtained by considering of the continuous-time signal FT, and sampling with., computing the corresponding (Inverse 12 $ $ as the spectrum
Energy (Parseval is called the energy spectrum ( energy spectral density : energy per radial Proof [ [ ] ] A similar property for power (see book seems less practical... 13
( ( $ $ ( A sufficient condition for the existence of the DTFT was that the signal is absolutely summable. But also for some other signals we can define the DTFT. Extension to signals with finite energy Signals with finite energy ( are not always absolutely summable (the reverse does hold:. Due to Parseval, the spectrum has equal energy: also finite. We can define a DTFT pair (signal/spectrum based on the Inverse DTFT (integral over a finite interval. Example: Ideal low-pass filter: elsewhere with copies every 14
[ has finite energy but is not absolutely summable (because ] converges to very slowly.35.3.25.2 ω =.35π 1.5 1 g[n].15.1.5 G(ω.5.5.1 2 15 1 5 5 1 15 2 n π π/2 π/2 π ω 15
Further extension to non-absolutely summable signals According to the equation, the Inverse DTFT of equals Hence ' ' % % This can be used to compute the DTFT of some signals which are not absolutely summable nor have finite energy (with impulses in the frequency domain, e.g., periodic signals. (constant signal is not absolutely summable. The DTFT is $ % Outside this interval: periodic (period, or. 16
has harmonically related frequencies: The DTFT of $ %! "! " [ ] is [ ] Outside this interval: periodic (period. More in general, consider [ ] for $ % (periodic outside this interval. A periodic signal, where the Fourier Series. (, with is the period (in samples. We obtain a line spectrum, just like with 17
% DTFT of a step The -transform of a unit step is ROC: The unit cicle is not in the ROC, thus the DTFT can only be defined in generalized sense. ( is not absolutely summable and does not have finite energy. Define the discrete-time sign function: sgn Then sgn sgn 18
sgn sgn sgn sgn sgn Proof (indication Using the Inverse DTFT: { } (trig. tricks... sgn Alternative proof Using the Fourier transform of sgn sgn : for For equals (in contrast to we consider the DC component of the function, which, which motivates why we looked at sng. has impulses at these frequencies. 19
Plot 5 4 real imag 3 2 1 1 2 3 4 5 π π/2 π/2 π The real-valued part 1 is constant, because we defined sgn instead of. Compare this to the Fourier transform of a continuous-time step function:. 2
Shift in time If! " and is a delay by samples, then so that! " "! The delay only affects the phase (which drops with a negative slope as function of. A linear phase term ( corresponds to a delay. 21
Shift in frequency If is a frequency shift of equals modulated by a complex exponential function by, then. Likewise: The modulation shifts the spectrum of to frequency. 22
Example of modulation with 1 x[n] = (.95 n 1 y[n] = (.95 n cos(2π n/1.5 5 5 n X(ω 4 2 π π/2 π/2 π ω 1 5 5 n 4 2 Y(ω π π/2 π/2 π ω 23
More generally: product of two signals The DTFT of the product! " is [ [ ] ] Hence: a product in time becomes a convolution in frequency domain (dual to the previous result Special case (seen before: because. 24
Real-valued signals For real-valued signals,. Hence and thus even in ; # # odd in It suffices to consider the spectrum on the interval $ $. Even real-valued signals If moreover, then is real-valued: The phase spectrum # is except for jumps of due to sign changes in. 25
and Summary of properties (cf Table 11.1 p.75 Parseval: (Note error in book regarding property (6, (7:. should be, 26
$ $ % $, or Discrete Fourier Transform (DFT Suppose has a finite length of samples (support is periodic with period, and we consider only 1 period. The DTFT is a continuous function of, with. We sample with samples: with We obtain -plane is called the Discrete Fourier Transform (DFT. 27
samples in frequency suffice to recover $ $ interval: periodic or zero (outside this Compututionally efficient due to the Fast Fourier Transform (FFT For a periodic with period, this corresponds to a Fourier Series. The DFT en its properties are discussed in EE2S31 (Q4, and in the practical of EE2T11 (Q3. 28
Relations interpolation continous-time periodic (1 period continuous-time sampling discrete-time discrete-time periodic (1 period windowing reconstruction (sinc-interpolation FS IFS FT IFT DTFT sampling windowing IDTFT DFT sampling IDFT windowing (LPF interpolation freq.discrete freq.continuous freq.continuous freq.discrete (line spectrum periodic (1 period Generally: periodic discrete short long product convolution 29