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Lecture 2 ELE 301: Signals and Systems Lecture 2 ELE 301: Signals and Systems Prof. Paul Cuff Princeton University Fall 2011-12 Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 1 / 70 Models of Continuous Time Signals Today's topics: Signals I Sinuoidal signals I Exponential signals I Complex exponential signals I Unit step and unit ramp I Impulse functions Systems I Memory I Invertibility I Causality I Stability I Time invariance I Linearity Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 2 / 70 Sinusoidal Signals A sinusoidal signal is of the form x(t) = cos(!t + θ): where the radian frequency is !, which has the units of radians/s. Also very commonly written as x(t) = A cos(2πft + θ): where f is the frequency in Hertz. We will often refer to ! as the frequency, but it must be kept in mind that it is really the radian frequency, and the frequency is actually f . Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 3 / 70 The period of the sinuoid is 1 2π T = = f ! with the units of seconds. The phase or phase angle of the signal is θ, given in radians. cos(ωt) cos(ωt θ) − 0 0 -2T -T T 2T t -2T -T T 2T t Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 4 / 70 Complex Sinusoids The Euler relation defines ejφ = cos φ + j sin φ. A complex sinusoid is Aej(!t+θ) = A cos(!t + θ) + jA sin(!t + θ): e jωt -2T -T 0 T 2T Real sinusoid can be represented as the real part of a complex sinusoid Aej(!t+θ) = A cos(!t + θ) <f g Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 5 / 70 Exponential Signals An exponential signal is given by x(t) = eσt If σ < 0 this is exponential decay. If σ > 0 this is exponential growth. 2 2 t t 1 e− 1 e -2 -1 0 1 2 t -2 -1 0 1 2 t Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 6 / 70 Damped or Growing Sinusoids A damped or growing sinusoid is given by x(t) = eσt cos(!t + θ) Exponential growth (σ > 0) or decay (σ < 0), modulated by a sinusoid. eσt cos(ωt) eσt cos(ωt) σ < 0 σ > 0 -2T -T 0 T 2T t -2T -T 0 T 2T t Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 7 / 70 Complex Exponential Signals A complex exponential signal is given by e(σ+j!)t+jθ = eσt (cos(!t + θ) + i sin(!t + θ)) A exponential growth or decay, modulated by a complex sinusoid. Includes all of the previous signals as special cases. e(σ+ jω)t e(σ+ jω)t σ < 0 σ > 0 -2T -T 0 T 2T t -2T -T 0 T 2T t Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 8 / 70 Complex Plane Each complex frequency s = σ + j! corresponds to a position in the complex plane. jω Left Half Plane Right Half Plane σ < 0 σ > 0 σ Decreasing Increasing Signals Signals Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 9 / 70 Demonstration Take a look at complex exponentials in 3-dimensions by using \TheComplexExponential" at demonstrations.wolfram.com Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 10 / 70 Unit Step Functions The unit step function u(t) is defined as 1; t 0 u(t) = ≥ 0; t < 0 Also known as the Heaviside step function. Alternate definitions of value exactly at zero, such as 1/2. 1 u(t) 2 1 0 1 2 t − − Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 11 / 70 Uses for the unit step: Extracting part of another signal. For example, the piecewise-defined signal e−t ; t 0 x(t) = ≥ 0; t < 0 can be written as x(t) = u(t)e−t 2 2 t t 1 u(t)e− 1 e− -2 -1 0 1 2 -2 -1 0 1 2 t t Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 12 / 70 Combinations of unit steps to create other signals. The offset rectangular signal 8 0; t 1 < ≥ x(t) = 1; 0 t < 1 ≤ : 0; t < 0 can be written as x(t) = u(t) u(t 1): − − 1 1 1 u(t) u(t 1) u(t) u(t 1) − − − 1 0 1 2 t 1 0 1 2 t 1 0 1 2 t − − − Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 13 / 70 Unit Rectangle Unit rectangle signal: 1 if t 1=2 rect(t) = j j ≤ 0 otherwise. 1 rect(t) t -1/2 0 1/2 Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 14 / 70 Unit Ramp The unit ramp is defined as t; t 0 r(t) = ≥ 0; t < 0 The unit ramp is the integral of the unit step, Z t r(t) = u(τ)dτ −∞ r(t) 1 2 1 0 1 2 t − − Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 15 / 70 Unit Triangle Unit Triangle Signal 1 t if t < 1 ∆(t) = − j j j j 0 otherwise. 1 Δ(t) t -1 0 1 Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 16 / 70 More Complex Signals Many more interesting signals can be made up by combining these elements. Example: Pulsed Doppler RF Waveform (we'll talk about this later!) τ cos(ωt) A 0 A − -2T -T 0 T 2T RF cosine gated on for τ µs, repeated every T µs, for a total of N pulses. Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 17 / 70 Start with a simple rect(t) pulse rect(t) 1 1/2 0 1/2 1 − − Scale to the correct duration and amplitude for one subpulse A A rect(t/τ) -2T -T 0 T 2T Combine shifted replicas 2 ∑ A rect((t nT)/τ) A n= 2 − − -2T -T 0 T 2T This is the envelope of the signal. Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 18 / 70 Then multiply by the RF carrier, shown below cos(ωt) 1 0 1 − -2T -T 0 T 2T to produce the pulsed Doppler waveform 2 ∑ A rect((t nT)/τ)cos(ωt) n= 2 − A − 0 A − -2T -T 0 T 2T Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 19 / 70 Impulsive signals (Dirac's) delta function or impulse δ is an idealization of a signal that is very large near t = 0 is very small away from t = 0 has integral 1 for example: 1/ε 1/ε ε t 2ε t the exact shape of the function doesn't matter is small (which depends on context) Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 20 / 70 On plots δ is shown as a solid arrow: δ(t) 1 0 1 t − t + 1 + δ(t) 1 0 1 t − Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 21 / 70 \Delta function" is not a function Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 22 / 70 Formal properties Formally we define δ by the property that Z 1 f (t)δ(t) dt = f (0) −∞ provided f is continuous at t = 0 idea: δ acts over a time interval very small, over which f (t) f (0) ≈ δ(t) is not really defined for any t, only its behavior in an integral. Conceptually δ(t) = 0 for t = 0, infinite at t = 0, but this doesn't 6 make sense mathematically. Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 23 / 70 Example: Model δ(t) as gn(t) = n rect(nt) as n . This has an area (n)(1=n) = 1. If f (t) is continuous at t = 0, ! 1 then Z 1 Z 1 Z 1 f (t)δ(t) dt = lim f (t)gn(t) dt = f (0) gn(t) dt = f (0) −∞ n!1 −∞ −∞ gn(t) 1 ε ε g2(t) g1(t) t 1 0 1 − gn(t) f (t) f (0) t 1 0 1 − Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 24 / 70 Scaled impulses αδ(t) is an impulse at time T , with magnitude or strength α We have Z 1 αδ(t)f (t) dt = αf (0) −∞ provided f is continuous at 0 On plots: write area next to the arrow, e.g., for 2δ(t), PSfrag replacements 2 t 0 Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 25 / 70 Multiplication of a Function by an Impulse Consider a function φ(x) multiplied by an impulse δ(t), φ(t)δ(t) If φ(t) is continuous at t = 0, can this be simplified? Substitute into the formal definition with a continuous f (t) and evaluate, Z 1 Z 1 f (t)[φ(t)δ(t)] dt = [f (t)φ(t)] δ(t) dt −∞ −∞ = f (0)φ(0) Hence φ(t)δ(t) = φ(0)δ(t) is a scaled impulse, with strength φ(0). Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 26 / 70 Sifting property The signal x(t) = δ(t T ) is an impulse function with impulse at − t = T . For f continuous at t = T , Z 1 f (t)δ(t T ) dt = f (T ) −∞ − Multiplying by a function f (t) by an impulse at time T and integrating, extracts the value of f (T ).
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