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This Lecture Covers the Following Topics ECE4330 Lecture 24 Discrete-Time Fourier Transform (DTFT): Application to LTID System Analysis and Digital Filter Design Prof. Mohamad Hassoun This Lecture covers the following topics Background: Connections with continuous-time systems The Sampling of 푓(푡) to obtain the sequence 푓[푘]: A brief review Fourier transform of discrete-time signals (DTFT) Connections between the DTFT and the Z-transform Discrete-time linear system analysis using the DTFT Response of a discrete-time system to an everlasting sinusoid Digital filter design based on analog filter approximations: o Impulse-invariance method: Numerical integration example o Impulse-invariance (direct) poles mapping method o The bilinear transformation Frequency domain-based digital filter design Matlab’s digital filter design tool: fdatool Nonlinear LSE-based digital filter design Numerical methods as recursive filters: Error analysis o Trapezoidal integration rule o Simpson’s 1/3 integration rule Appendix: General summary Background: Connections with Continuous-Time Systems In previous lectures we have introduced the Fourier transform that allowed us to analytically represent a continuous-time (CT) aperiodic signal in terms of the radian frequency, 휔, as +∞ 푓(푡) ↔ 퐹(휔) = ∫ 푓(푡)푒−푗휔푡푑푡 −∞ We will refer to this transformation as the CTFT. For example, we have derived earlier the CTFT pair, 1 푓(푡) = 푒−푎푡푢(푡) ↔ 퐹(휔) = (푎 > 1) 푎 + 푗휔 Tables of Fourier transform pairs and Fourier transform properties are available. The Fourier transform was also applied to a stable, continuous- time linear system which allowed us to transform the system’s linear differential equation representation (with zero initial state) into a rational function representation (frequency response function) of the form, 퐾(푗휔 − 푧 )(푗휔 − 푧 ) … (푗휔 − 푧 ) 퐻(휔) = 1 2 푚 (푗휔 − 훾1)(푗휔 − 훾2) … (푗휔 − 훾푛) where the 푧푖 and the 훾푖 are the zeros and poles of 퐻(휔), respectively. This rational function is the Fourier transform of the system’s unit-impulse response, ℎ(푡) ↔ 퐻(휔). The frequency response 퐻(휔) can also be obtained (for stable linear systems) by simply making the substituting 푠 = 푗휔 in the expression for the system’s transfer function, 퐻(푠). Given a causal input signal 푓(푡)푢(푡), with Fourier transform 퐹(휔), the zero-state frequency-domain response, 푌푧푠(휔), of a stable linear system with frequency response 퐻(휔) is given by, 푌푧푠(휔) = 퐻(휔)퐹(휔) with a corresponding time-domain zero-state response given by −1 푦푧푠(푡) = 퐹 {퐻(휔)퐹(휔)} If the input signal 푓(푡) is the everlasting sinusoid 푓(푡) = 퐶cos(휔0푡 + 휃) [푓(푡) = 퐶0 is a special case where 휔0 = 0] or, more generally, periodic with Fourier series ∞ 푓(푡) = 퐶0 + ∑ 퐶푛 cos(푛휔0푡 + 휃푛) 푛=1 then, the zero-state response becomes the steady-state response, 푦푧푠(푡) = 푦푠푠(푡). The steady-state response can be obtained using the formula ∞ 푦푠푠(푡) = 퐻(0)퐶0 + ∑|퐻(푛휔0)|퐶푛 cos[푛휔0푡 + ∠퐻(푛휔0) + 휃푛] 푛=1 This lecture answers the following questions in connection with a discrete- time signal 푓[푘] and a stable LTDI system with impulse response ℎ[푘] (or, equivalently, a stable system with a known LTI difference equation): Is there a discrete-time Fourier transform (DTFT) for 푓[푘]? How can one obtain the DTFT for 푓[푘]? How can the inverse transformation be obtained? What are the properties of the DTFT? How can one obtain the system’s frequency response function? How can one obtain the zero-state and steady-state responses? How can we design a discrete-time filter that is equivalent to a continuous-time (analog) filter? The following is a summary of the three transformation methods that we had utilized in this course along with the DTFT that will be covered in the rest of this lectures. Refer to the appendix for an extended summary. The Sampling of 푓(푡) to Obtain the Sequence 푓[푘]: A Brief Review The discrete-time signal (or sequence) 푓[푘] will be assumed to have been obtained by uniformly sampling a continuous-time signal 푓(푡) at uniform intervals of duration 푇푠, or with a corresponding sampling frequency 푓푠 = 1 . We will assume that the sampling frequency satisfies the (equivalent) 푇푠 inequalities 1 푓 > 2퐵, 휔 > 4휋퐵 and 푇 < 푠 푠 푠 2퐵 where 퐵 is the effective bandwidth (in Hz) of the signal 푓(푡). (Refer to Lectures 18 and 23 for bandwidth and sampling rate, respectively.) For example, for the bandlimited signal 푓(푡) = cos (휔0푡) the sampled signal is obtained by a (discretizing) variable transformation, 푡 → 푇푠푘, to ( ) ( )| ( ) ( ) obtain 푓 푇푠푘 = 푓 푡 푡=푇푠푘 = cos 휔0푇푠푘 = cos Ω0푘 , where we define Ω0 = 푇푠휔0 and 푘 = ⋯ , −2, −1, 0, 1, 2, … . We will refer to Ω0 as the discrete-time angular frequency, measured in rad/sample. Since for a pure 휔0 휋 sinusoid the bandwidth is 퐵 = , we must employ 푇푠 < . Now, we may 2휋 휔0 generate the sequence 푓[푘] by setting its values equal to 푓(푇푠푘). Example. Find a proper sampled signal 푓[푘] for the continuous-time signal 휋 푓(푡) = 3 cos (1500푡 + ) 3 1500 1 This sinusoid has a bandwidth 퐵 = . Any 푇 satisfying 푇 < = 2휋 푠 푠 2퐵 휋 ≅ 0.0021 would allow for adequate reconstruction (by the Sampling 1500 Theorem). So, we can set 푇푠 = 0.001 sec (푓푠 = 1000Hz) to arrive at the sampled signal 휋 휋 푓[푘] = 3 cos (1500(0.001)푘 + ) = 3 cos (1.5푘 + ) 3 3 Notice how the sampling rate 푇푠 = 0.001 leads to a scaled the frequency, 휔0 휔0 = 1500 rad/sec → Ω0 = 푇푠휔0 = = 1.5 rad/sample 푓푠 휋 Your turn: Plot 푓[푘] = 3 cos (1.5푘 + ) and compare it to 푓(푡). 3 We should avoid making 푇푠 unnecessarily small because that would lead to a reduction in speed when digital processors are used to process the sampled signal. It should be emphasized that once the samples are taken from 푓(푡) and stored in memory (in digital/quantized form), the time scale is lost. The discrete-time signal is just a sequence of numbers, and these numbers carry no information about the sampling period, 푇푠. Therefore, the process that generates the discrete-time sequence, 푓[푘], from the continuous-time signal, 푓(푡), performs two operations: 1. It samples the signal 푓(푡) every 푇푠 seconds and assigns the value 푓(푇푠푘) to the 푘th sample of the sequence, 푓[푘]. 2. It scales (normalizes) the time axis by 푘 = 푡/푇푠, so that the distance between successive samples is always unity. Thus, it changes the time axis from 푡 to its normalized integer time, 푘. In Lecture 23, we showed that the CT Fourier transform of the sampled signal 푓(푇푠푘) is, ∞ ∞ 1 휔 퐺(휔) = 퐹(휔) ∗ [휔 ∑ 훿(휔 − 푘휔 )] = 푠 ∑ 퐹(휔 − 푘휔 ) 2휋 푠 푠 2휋 푠 푘=−∞ 푘=−∞ In the above expression, 퐹(휔) is the Fourier transform of 푓(푡) and 휔푠 = 2휋 is the radian sampling frequency. The expression can be rewritten as, 푇푠 ∞ 퐺(푓) = 푓푠 ∑ 퐹(푓 − 푘푓푠) 푘=−∞ Therefore, the process that generates the discrete-time sequence 푓[푘] from the continuous-time signal 푓(푡) can be viewed, in the frequency domain, as performing the following operations: 1. It scales the spectrum of 푓(푡) by 푓푠. 2. It duplicates the scaled spectrum at all integer multiples of the sampling frequency, 푘푓푠. Fourier Transform of Discrete-Time Signals (DTFT) In analogy to using the Fourier transform to arrive at a frequency-domain characterization of a continuous signal, 푓(푡), we define the discrete-time Fourier transform (DTFT) of the sampled signal (sequence) 푓[푘] as the infinite sum ∞ 퐹(Ω) = ∑ 푓[푘]푒−푗Ω푘 푘=−∞ 휔 where Ω = 푇푠휔 = is the discrete-time angular frequency (measured in 푓푠 rad/sample). As with other transforms, we will describe this relationship between the discrete-time signal and its DTFT as a transform pair, 푓[푘] ↔ 퐹(Ω) It is interesting to note how one can obtain the DTFT of a sampled signal from the (continuous-time) Fourier transform of 푓(푡), as shown next. As we did in the previous lecture, the sampled signal 푓[푘] is obtained by multiplying 푓(푡) by the impulse train 훿푠(푡), ∞ 푓(푡)훿푠(푡) = 푓(푡) ∑ 훿(푡 − 푇푠푘) 푘=−∞ Applying the Fourier transformation to the above sampled signal we get +∞ +∞ ∞ −푗휔푡 −푗휔푡 ∫ 푓(푡)훿푠(푡)푒 푑푡 = ∫ [푓(푡) ∑ 훿(푡 − 푇푠푘)] 푒 푑푡 −∞ −∞ 푘=−∞ ∞ +∞ ∞ −푗휔푡 −푗휔푇푠푘 = ∑ ∫ 푓(푡)푒 훿(푡 − 푇푠푘)푑푡 = ∑ 푓(푇푠푘)푒 푘=−∞ −∞ 푘=−∞ Upon defining Ω = 푇푠휔 and replacing 푓(푇푠푘) by its sample values 푓[푘], in the above expression we arrive at the DTFT, ∞ 퐹(Ω) = ∑ 푓[푘]푒−푗Ω푘 푘=−∞ Example. Find the DTFT of the causal signal 푓[푘] = 훾푘푢[푘] and plot its spectra for 훾 = 0.8. ∞ ∞ ∞ 푘 퐹(Ω) = ∑ 훾푘푢[푘]푒−푗Ω푘 = ∑ 훾푘푒−푗Ω푘 = ∑(훾푒−푗Ω) 푘=−∞ 푘=0 푘=0 is the sum of a geometric series with common ratio 훾푒−푗Ω. The sum converges to 1 푒푗Ω 퐹(Ω) = = 1 − 훾푒−푗Ω 푒푗Ω − 훾 if |훾푒−푗Ω| < 1, but since |푒−푗Ω| = 1, then |훾| < 1 is required for the transform to exist. Therefore, we arrive at our first DTFT pair, 1 푒푗Ω 훾푘푢[푘] ↔ = |훾| < 1 1 − 훾푒−푗Ω 푒푗Ω − 훾 Setting 훾 = 0.8 in the DTFT we obtain 1 퐹(Ω) = 1 − 0.8푒−푗Ω or, equivalently, by employing Euler’s identity we may write 1 퐹(Ω) = 1 − 0.8[cos(Ω) − 푗sin(Ω)] The magnitude and angle responses are then (your turn: derive them) 1 |퐹(Ω)| = √1 + 0.64 − 1.6 cos(Ω) 0.8 sin(Ω) angle(퐹(Ω)) = − tan−1 ( ) 1 − 0.8 cos(Ω) The following two plots show the spectra of 퐹(Ω).
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