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Roadmap 11/30/2012 8:19 AM1. Response of LTI Systems2. Signal Distortion in Transmission3. Transmission Loss and Decibels4. Filters and Filtering5. Quadrature Filters and Hilbert Transforms6. Correlation and Spectral Density 2
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11/30/2012 8:19 AMRESPONSE OF LTI SYSTEMS• Impulse Response and the Superposition Integral• Transfer Functions and Frequency Response• Block-Diagram Analysis 3
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Impulse Response and theSuperposition Integral 11/30/2012 8:19 AMThe output y(t) is then the forced response due entirely to x(t) where F[x(t)] stands for the functional relationship between input and output 4
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What is LTI means ? The linear property means that the system equation obeys the principle of 11/30/2012 8:19 AM superposition. Thus, if where ak are constants, thenThe time-invariance property means that the system’s characteristics remain fixedwith time. Thus, a time-shifted input x(t – td) produces 5 so the output is time-shifted but otherwise unchanged.
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Direct analysis of a lumped-parameter system starting with the element equationsleads to the input–output relation as a linear differential equation in the form 11/30/2012 8:19 AMUnfortunately, this Eq. doesn’t provide us with a direct expression for y(t)To obtain an explicit input–output equation, we must first define the system’simpulse response function which equals the forced response when x(t) = δ(t) 6
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Any continuous input signal can be written as the convolution x(t) = x(t)*δ(t) ,so 11/30/2012 8:19 AMFrom the time-invariance property, F[δ(t - λ)] = h(t – λ) and hence 7 superposition integral
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Various techniques exist for determining h(t) from a differential equation or someother system model.However, you may be more comfortable taking x(t) = u(t) and calculating the 11/30/2012 8:19 AMsystem’s step response from which 8
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EXAMPLE: Time Response of a First-Order System 11/30/2012 8:19 AM This circuit is a first-order system governed by the differential equationFrom either the differential equation or the circuit diagram, the step response isreadily found to beInterpreted physically, the capacitor starts atzero initial voltage and charges toward y(∞) = 1 9with time constant RC when x(t) = u (t)
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The corresponding impulse response 11/30/2012 8:19 AMThe response to an arbitrary input x(t) can now be found by putting the impulseresponse equation in the superposition integral.Rectangular pulse applied at t = 0, so x(t) = A for 0 < t < τ .The convolution y(t) = h(t) * x(t) divides into three parts, with the result that 10
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Transfer Functions and Frequency Response 11/30/2012 8:19 AMTime-domain analysis becomes increasingly difficult for higher-order systems, wegot a clearer view of system response by going to the frequency domain.As a first step in this direction, we define the system transfer function to be theFourier transform of the impulse response, namely,This definition requires that H(f) exists, at least in a limiting sense. In the case ofan unstable system, h(t) grows with time and H(f) does not exist. 12
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When h(t) is a real time function, H(f) has the hermitian symmetry 11/30/2012 8:19 AM 13
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The steady-state forced response is 11/30/2012 8:19 AMConverting H(f0) to polar form then yields 14
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if 11/30/2012 8:19 AMthenSince Ay/Ax = |H(f0)| at any frequency f0,|H(f)| represents the system’s amplitude ratio as a function of frequency(sometimes called the amplitude response or gain)arg H(f) represents the phase shift, since φy – φx = arg H(f) 15Plots of |H(f)| and arg H(f) versus frequency give the system’s frequency response
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let x(t) be any signal with spectrum X(f) we take the transform of y(t) = x(t) * h(t) to obtain 11/30/2012 8:19 AM The output spectrum Y(f) equals the input spectrum X(f) multiplied by the transfer function H(f).If x(t) is an energy signal, then y(t) will be an energy signal whose spectral densityand total energy are given by 16
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11/30/2012 8:19 AMOther ways of determining H(f)calculate a system’s steady-state phasor response, 17
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EXAMPLE: Frequency Response of a First-Order System 11/30/2012 8:19 AMy(t)/x(t) = ZC/(ZC + ZR) when x(t) = ejωt ZR = R and ZC = 1/jωC 18
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11/30/2012 8:19 AMWe call this particular system a lowpass filter because it has almost no effecton the amplitude of low-frequency components, say |f| << B , while itdrastically reduces the amplitude of high-frequency components, say |f| << B 19The parameter B serves as a measure of the filter’s passband or bandwidth.
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If W << B 11/30/2012 8:19 AM |H(f)| ≈ 1, and arg H(f) ≈ 0 over the signal’s frequency range |f| < W Thus, Y(f) = H(f)X(f) ≈ X(f) and y(t) ≈ x(t)so we have undistorted transmission through the filter. 20
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If W ≈ B 11/30/2012 8:19 AM Y(f) depends on both H(f) and X(f).We can say that the output is distorted, since y(t)will differ significantly from x(t), but time-domaincalculations would be required to find the actualwaveform. 21
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If W >> B 11/30/2012 8:19 AMThe input spectrum has a nearly constantvalue X(0) for |f| < B Y(f) ≈ X(0)H(f), y(t) ≈ X(0)h(t)The output signal now looks like the filter’simpulse response. Under this condition, wecan reasonably model the input signal as animpulse. 22
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11/30/2012 8:19 AMOur previous time-domain analysis with a rectangular input pulseconfirms these conclusions since the nominal spectral width of the pulseis W = 1/τ.The case W << B thus corresponds to 1/τ << 1/2πRC or τ/RC >> 1, andy(t) ≈ x(t).Conversely, W >> B corresponds to τ/RC << 1 where y(t) looks more likex(t). 23
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Block-Diagram Analysis 11/30/2012 8:19 AMWhen the subsystems in question are described by individual transferfunctions, it is possible and desirable to lump them together and speak of theoverall system transfer function. 24
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EXAMPLE: Zero-Order Hold 11/30/2012 8:19 AM 26
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To confirm this result by another route,let’s calculate the impulse response h(t) drawing upon the definition that 11/30/2012 8:19 AMy(t) = h(t) when x(t) = δ(t)The input to the integrator then is x(t) - x(t - T) = δ(t) - δ(t - T), soWhich represents a rectangular pulse starting at t = 0. Rewriting the impulseresponse as h(t) = ∏ [(t – T/2)/T] helps verify the transform relationh(t) ↔ H(f). 27
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