Linear Time-Invariant (LTI) Systems Continuous-Time (CT) systems process CT signals. Discrete-Time (DT) systems process DT signals. 푓(푡) represents a continuous-time signal. 푓[푘] represents a discrete-time signal. Bandwidth of 푓(푡) is 퐵 (Hz). B is determined by applying the Fourier transform. Sampled signal: 푓(푡) → 푓[푘] = 푓(푇푠푘), with integer 푘. 1 Sampling Theorem: 푇 < is required for perfect reconstruction of 푓(푡) from 푓[푘]. 푠 2퐵 LTICT system dynamics are governed by a linear differential equation which relates the system response 푦(푡) to the input signal 푓(푡), subject to certain initial conditions. LTIDT system dynamics are governed by a linear difference equation which relates the system response 푦[푘] to the input signal 푓[푘], subject to certain initial conditions. System analysis methods allow us to solve for 푦 = 푦푧푖 + 푦푧푠, where 푦푧푖 is the (zero-input) response due to initial conditions (or energy stored in the system) and 푦푧푠 is the (zero-state) response due to the input 푓 (with stored energy set to zero). System Transfer functions are rational functions (ratio of two polynomials) that are obtained using the Laplace transform for LTICT systems, and using the Z-transform for LTIDT systems. 푁(푠) LTICT system transfer function: 퐻(푠) = [퐷(푠) is the characteristic equation of the system]. 퐷(푠) The solutions, 푠푖, of 퐷(푠) = 0 are the system poles or natural frequencies (generally complex) 푠푖푡 LTICT system natural modes take the form, 푒 . Stability requires 푅푒(푠푖) < 0 for all 푠푖. 푁(푧) LTIDT system transfer function: 퐻(푧) = [퐷(푧) is the characteristic equation of the system] 퐷(푧) The solutions, 훾푖, of 퐷(푧) = 0 are the system poles or natural frequencies (generally complex). 푘 LTIDT system natural modes take the form, 훾푖 . Stability requires |훾푖| < 1 for all 훾푖. Fourier Series: The Fourier series allows us to express a periodic signal 푓(푡), with a fundamental frequency 휔표, as the sum of a constant (signal average) and an infinite number of harmonic sinusoids: ∞ Compact trigonometric Fourier series: 푓(푡) = 퐶0 + ∑푛=1 퐶푛 cos(푛휔표푡 + 휃푛) +∞ 푗푛휔0푡 Exponential Fourier series: 푓(푡) = ∑푛=−∞ 퐷푛푒 1 2 2 1 ∞ 2 2 +∞ 2 Parseval’s theorem: ∫ 푓 (푡)푑푡 = 퐶0 + ∑푛=1 퐶푛 = 퐷0 + 2 ∑푛=1|퐷푛| 푇0 푇0 2 Laplace Transform: ∞ 퐹(푠) = ∫ 푓(푡)푒−푠푡 푑푡 0− Transforms a CT signal, 푓(푡), into a complex signal, 퐹(푠), in the 푠-domain (푠 = 휎 + 푗휔). Table of Laplace transform pairs 푓(푡) ↔ 퐹(푠) and Laplace transform properties are available that allow us to find 퐹(푠) from 푓(푡) and 푓(푡) from 퐹(푠), and avoid using the above integral. The Laplace transform simplifies the analysis of a LTICT system by transforming its linear differential equation into an algebraic equation: 푄(푠)푌(푠) + 퐼(푠) = 푃(푠)퐹(푠), where 푦(푡) ↔ 푌(푠) and 푓(푡) ↔ 퐹(푠). Here, 푄(푠), 푃(푠) and 퐼(푠) are polynomials. 퐼(푠) depends only on the system’s initial conditions. 푌(푠) 푃(푠) Such algebraic equation can then be solved to obtained the transfer function 퐻(푠) = = , where 퐹(푠) 푄(푠) 퐼(푠) is set to zero (i.e., initial conditions are set to zero). The system’s unit-impulse response, ℎ(푡), and the transfer function, 퐻(푠), form a Laplace transform pair: ℎ(푡) ↔ 퐻(푠) The Laplace transform can be used to analyze stable and unstable LTICT systems. It can generate the 퐼(푠) 푃(푠) response: 푌(푠) = 푌 (푠) + 푌 (푠), where 푌 (푠) = − and 푌 (푠) = 퐹(푠) = 퐻(푠)퐹(푠) 푧푖 푧푠 푧푖 푄(푠) 푧푠 푄(푠) 퐼(푠) Performing the inverse Laplace transformation on 푌 (푠) = − and 푌 (푠) = 퐻(푠)퐹(푠) (which 푧푖 푄(푠) 푧푠 involves applying partial fraction expansion and the use of the Laplace transform Table) lead to the desired time-domain solution: 퐼(푠) 푦(푡) = 푦 (푡) + 푦 (푡) = 퐿−1 {− } + 퐿−1{퐻(푠)퐹(푠)} 푧푖 푧푠 푄(푠) In addition to the analysis of LTICT systems, the Laplace transform finds important applications in automatic control. Fourier Transform: +∞ 퐹(휔) = ∫ 푓(푡)푒−푗휔푡푑푡 −∞ Transforms a CT signal, 푓(푡), into a complex signal, 퐹(휔), in the 휔-domain. Tables of Fourier transform pairs, 푓(푡) ↔ 퐹(휔), and Fourier transform properties are available to assist in obtaining the transform and its inverse. ∞ If a causal signal 푓(푡) is absolutely integrable (i.e., |푓(푡)|푑푡 < ∞), and has a Laplace transform ∫0− 퐹(푠), then (and only then) the Fourier transform of 푓(푡) can be obtained from 퐹(푠) as 퐹(휔) = 퐹(푠)|푠=푗휔 Similarly, the frequency response function 퐻(휔) of a stable LTICT system can be obtained from the transfer function 퐻(푠) by simply replacing 푠 by 푗휔. The Fourier transform cannot handle initial conditions like the Laplace transform. Therefore, it cannot be used to obtain the 푦푧푖(푡) component of the response. However, if the system is stable, the Fourier transform can be used to obtain the zero-state response according to the formula: 푦푧푠(푡) = 퐹−1{퐻(휔)퐹(휔)}. Here, the notation 퐹−1{} stands for the inverse Fourier transform. If 푓(푡) is a constant, an everlasting sinusoid or (more generally) an everlasting periodic signal then −1 퐹 {퐻(휔)퐹(휔)} gives the steady-state response, 푦푠푠(푡). The response of a stable system to a periodic 푓(푡), that is expressed in terms of its trigonometric ∞ Fourier series, 푓(푡) = 퐶0 + ∑푛=1 퐶푛 cos(푛휔표푡 + 휃푛), is given by the formula ∞ 푦푠푠(푡) = 퐻(0)퐶0 + ∑|퐻(푛휔0)|퐶푛 cos[푛휔0푡 + ∠퐻(푛휔0) + 휃푛] 푛=1 +∞ 푗푛휔0푡 For a periodic 푓(푡) that is expressed in terms of its exponential Fourier series 푓(푡) = ∑푛=−∞ 퐷푛푒 the steady-state response is determined using the formula, +∞ +∞ 푗푛휔0푡 푗[푛휔0푡+∠퐻(푛휔0)+∠퐷푛] 푦푠푠(푡) = ∑ 퐻(푛휔0)퐷푛푒 = ∑ |퐻(푛휔0)||퐷푛|푒 푛=−∞ 푛=−∞ The Fourier transform is very useful in many applications including the analysis and design of communication systems, design of analog filters, sampling theory, and in determining a signal’s bandwidth. We can express the bandwidth equation for a signal 푓(푡) as (훼 is typically set to a value between 0.95 and 0.99). 휔 퐵|퐹(휔)|2푑휔 ∫0 ∞ = 훼 |퐹(휔)|2푑휔 ∫0 The above equation can be solved (analytically or numerically) for 휔퐵 from which we define the signal’s effective bandwidth (in Hz) as 휔 퐵 = 퐵 Hz 2휋 The Z-Transform: ∞ 퐹(푧) = ∑ 푓[푘]푧−푘 푘=−∞ Transforms a DT signal, 푓[푘], into a complex signal, 퐹(푧), in the 푧-domain (푧 is complex) Tables of Z-transform pairs, 푓[푘] ↔ 퐹(푧), and Z-transform properties are available to assist in obtaining the transform and its inverse. The Z-transform simplifies the analysis of a LTIDT system by transforming its linear difference equation into an algebraic equation: 푄(푧)푌(푧) + 퐼(푧) = 푃(푧)퐹(푧), where 푦[푘] ↔ 푌(푧) and 푓[푘] ↔ 퐹(푧). Here, 푄(푧), 푃(푧) and 퐼(푧) are polynomials. 퐼(푧) depends only on the system’s initial conditions. 푌(푧) 푃(푧) Such algebraic equation can then be solved to obtained the transfer function 퐻(푧) = = , where 퐹(푧) 푄(푧) 퐼(푧) is set to zero (i.e., initial conditions are set to zero). The system’s unit-impulse response, ℎ[푘], and the transfer function, 퐻(푧), form a Z-transform pair: ℎ[푘] ↔ 퐻(푧) The Z-transform can be used to analyze stable and unstable LTIDT systems. It can generate the 퐼(푧) 푃(푧) response: 푌(푧) = 푌 (푧) + 푌 (푧), where 푌 (푧) = − and 푌 (푧) = 퐹(푧) = 퐻(푧)퐹(푧) 푧푖 푧푠 푧푖 푄(푧) 푧푠 푄(푧) 퐼(푧) Performing the inverse Z-transformation on 푌 (푧) = − and 푌 (푧) = 퐻(푧)퐹(푧) (which involves 푧푖 푄(푧) 푧푠 applying partial fraction expansion and the use of the Z-transform Table) lead to the desired time- domain solution: 퐼(푧) 푦[푘] = 푦 [푘] + 푦 [푘] = 푍−1 {− } + 푍−1{퐻(푧)퐹(푧)} 푧푖 푧푠 푄(푧) Discrete-Time Fourier Transform (DTFT): ∞ 퐹(Ω) = ∑ 푓[푘]푒−푗Ω푘 푘=−∞ 휔 where Ω = 푇푠휔 = is the discrete-time angular frequency (measured in rad/sample), and 푓푠 is the sampling 푓푠 frequency. The DTFT transforms a DT signal, 푓[푘], into a complex signal, 퐹(Ω), in the Ω-domain. Tables of DTFT pairs, 푓[푘] ↔ 퐹(Ω), and DTFT properties are available to assist in obtaining the transform and its inverse. ∞ If a causal signal 푓[푘] is absolutely summable (i.e., ∑푘=−∞|푓[푘]| < ∞), and has a Z-transform 퐹(푧), then (and only then) the DTFT of 푓[푘] can be obtained from 퐹(푧) as 퐹(Ω) = 퐹(푧)|푧=푒푗Ω The frequency response function 퐻(Ω) of a stable LTIDT system can be obtained from the transfer function 퐻(푧) by simply replacing 푧 by 푒푗Ω. The DTFT relationship to the Z-transform is very similar to that of the relationship of the (CT) Fourier transform to the Laplace transform. The DTFT cannot handle initial conditions like the Z-transform. Therefore, it cannot be used to obtain the 푦푧푖[푘] component of the response. However, if the system is stable, the DTFT can be used to −1 obtain the zero-state response according to the formula: 푦푧푠[푘] = 퐹 {퐻(Ω)퐹(Ω)}.