S. Boyd EE102Lecture 1Signals• notation and meaning• common signals• size of a signal• qualitative properties of signals• impulsive signals1–1Signalsa signal is a function of time, e.g.,• f is the force on some mass• voutis the output voltage of some circuit• p is the acoustic pressure at some pointnotation:• f, vout, p or f(·), vout(·), p(·) refer to the whole signal or function• f(t), vout(1.2), p(t + 2) refer to the value of the signals at times t, 1.2,and t + 2, respectivelyfor times we usually use symbols like t, τ, t1, . . .Signals 1–2Example−1 0 1 2 3−101PSfrag replacementsp(t) (Pa)t (msec)Signals 1–3Domain of a signaldomain of a signal: t’s for which it is definedsome common domains:• all t, i.e., R• nonnegative t: t ≥ 0(here t = 0 just means some starting time of interest)• t in some interval: a ≤ t ≤ b• t at uniformly sampled points: t = kh + t0, k = 0, ±1, ±2, . . .• discrete-time signals are defined for integer t, i.e., t = 0, ±1, ±2, . . .(here t means sample time or epoch, not real time in seconds)we’ll usually study signals defined on all reals, or for nonnegative realsSignals 1–4Dimension & units of a signaldimension or type of a signal u, e.g.,• real-valued or scalar signal: u(t) is a real number (scalar)• vector signal: u(t) is a vector of some dimension• binary signal: u(t) is either 0 or 1we’ll usually encounter scalar signalsexample: a vector-valued signalv =v1v2v3might give the voltage at three places on an antennaphysical units of a signal, e.g., V, mA, m/secsometimes the physical units are 1 (i.e., unitless) or unspecifiedSignals 1–5Common signals with names• a constant (or static or DC) signal: u(t) = a, where a is some constant• the unit step signal (sometimes denoted 1(t) or U(t)),u(t) = 0 for t < 0, u(t) = 1 for t ≥ 0• the unit ramp signal,u(t) = 0 for t < 0, u(t) = t for t ≥ 0• a rectangular pulse signal,u(t) = 1 for a ≤ t ≤ b, u(t) = 0 otherwise• a sinusoidal signal:u(t) = a cos(ωt + φ)a, b, ω, φ are called signal parametersSignals 1–6Real signalsmost real signals, e.g.,• AM radio signal• FM radio signal• cable TV signal• audio signal• NTSC video signal• 10BT ethernet signal• telephone signalaren’t given by mathematical formulas, but they do have definingcharacteristicsSignals 1–7Measuring the size of a signalsize of a signal u is measured in many waysfor example, if u(t) is defined for t ≥ 0:• integral square (or total energy):Z∞0u(t)2dt• squareroot of total energy• integral-absolute value:Z∞0|u(t)| dt• peak or maximum absolute value of a signal: maxt≥0|u(t)|• root-mean-square (RMS) value:ÃlimT →∞1TZT0u(t)2dt!1/2• average-absolute (AA) value: limT →∞1TZT0|u(t)| dtfor some signals these measures can be infinite, or undefinedSignals 1–8example: for a sinusoid u(t) = a cos(ωt + φ) for t ≥ 0• the peak is |a|• the RMS value is |a|/√2 ≈ 0.707|a|• the AA value is |a|2/π ≈ 0.636|a|• the integral square and integral absolute values are ∞the deviation between two signals u and v can be found as the size of thedifference, e.g., RMS(u − v)Signals 1–9Qualitative properties of signals• u decays if u(t) → 0 as t → ∞• u converges if u(t) → a as t → ∞ (a is some constant)• u is bounded if its peak is finite• u is unbounded or blows up if its peak is infinite• u is periodic if for some T > 0, u(t + T ) = u(t) holds for all tin practice we are interested in more specific quantitative questions, e.g.,• how fast does u decay or converge?• how large is the peak of u?Signals 1–10Impulsive 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 1for example:PSfrag replacementst²1/²t = 0PSfrag replacementst2²1/²t = 0• the exact shape of the function doesn’t matter• ² is small (which depends on context)Signals 1–11on plots δ is shown as a solid arrow:PSfrag replacementstt = 0f(t) = δ(t)PSfrag replacementstt = 0t = −1f(t) = t + 1 + δ(t)Signals 1–12Formal propertiesformally we define δ by the property thatZbaf(t)δ(t) dt = f(0)provided a < 0, b > 0, and f is continuous at t = 0idea: δ acts over a time interval very small, over which f(t) ≈ f(0)• δ(t) = 0 for t 6= 0• δ(0) isn’t really defined•Zbaδ(t) dt = 1 if a < 0 and b > 0•Zbaδ(t) dt = 0 if a > 0 or b < 0Signals 1–13Zbaδ(t) dt = 0 is ambiguous if a = 0 or b = 0our convention: to avoid confusion we use limits such as a− or b+ todenote whether we include the impulse or notfor example,Z10+δ(t) dt = 0,Z10−δ(t) dt = 1,Z0−−1δ(t) dt = 0,Z0+−1δ(t) dt = 1Signals 1–14Scaled impulsesαδ(t − T ) is sometimes called an impulse at time T , with magnitude αwe haveZbaαδ(t − T )f(t) dt = αf(T )provided a < T < b and f is continuous at Ton plots: write magnitude next to the arrow, e.g., for 2δ,PSfrag replacementst02Signals 1–15Sifting propertythe signal u(t) = δ(t − T ) is an impulse function with impulse at t = Tfor a < T < b, and f continuous at t = T , we haveZbaf(t)δ(t − T ) dt = f(T )example:Z3−2f(t)(2 + δ(t + 1) − 3δ(t − 1) + 2δ(t + 3)) dt= 2Z3−2f(t) dt +Z3−2f(t)δ(t + 1) dt − 3Z3−2f(t)δ(t − 1) dt+ 2Z3−2f(t)δ(t + 3)) dt= 2Z3−2f(t) dt + f(−1) − 3f(1)Signals 1–16Physical interpretationimpulse functions are used to model physical signals• that act over short time intervals• whose effect depends on integral of signalexample: hammer blow, or bat hitting ball, at t = 2• force f acts on mass m between t = 1.999 sec and t = 2.001 sec•Z2.0011.999f(t) dt = I (mechanical impulse, N · sec)• blow induces change in velocity ofv(2.001) − v(1.999) =1mZ2.0011.999f(τ) dτ = I/mfor (most) applications we can model force as an impulse, at t = 2, withmagnitude ISignals 1–17example: rapid charging of capacitorPSfrag replacements1Vv(t)i(t)t = 01Fassuming v(0) = 0, what is v(t), i(t) for t > 0?• i(t) is very large, for a very short time• a unit charge is transferred to the capacitor ‘almost instantaneously’• v(t) increases to v(t) = 1 ‘almost instantaneously’to calculate i, v, we need a more detailed modelSignals 1–18for example, include small resistancePSfrag replacements1Vv(t)Rv(t)i(t)t = 01Fi(t) =dv(t)dt=1 − v(t)R, v(0) = 0PSfrag replacementsR1v(t) = 1 − e−t/RPSfrag replacementsR1/Ri(t) = e−t/R/Ras R →