Introduction and types of signals and its charachteristics
- 1. S. Boyd EE102 Lecture 1 Signals • notation and meaning • common signals • size of a signal • qualitative properties of signals • impulsive signals 1–1
- 2. Signals a signal is a function of time, e.g., • f is the force on some mass • vout is the output voltage of some circuit • p is the acoustic pressure at some point notation: • 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, respectively for times we usually use symbols like t, τ, t1, . . . Signals 1–2
- 3. Example −1 0 1 2 3 −1 0 1 PSfrag replacements p(t) (Pa) t (msec) Signals 1–3
- 4. Domain of a signal domain of a signal: t’s for which it is defined some 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 reals Signals 1–4
- 5. Dimension & units of a signal dimension 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 1 we’ll usually encounter scalar signals example: a vector-valued signal v = v1 v2 v3 might give the voltage at three places on an antenna physical units of a signal, e.g., V, mA, m/sec sometimes the physical units are 1 (i.e., unitless) or unspecified Signals 1–5
- 6. Common 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 parameters Signals 1–6
- 7. Real signals most real signals, e.g., • AM radio signal • FM radio signal • cable TV signal • audio signal • NTSC video signal • 10BT ethernet signal • telephone signal aren’t given by mathematical formulas, but they do have defining characteristics Signals 1–7
- 8. Measuring the size of a signal size of a signal u is measured in many ways for example, if u(t) is defined for t ≥ 0: • integral square (or total energy): Z ∞ 0 u(t)2 dt • 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: Ã lim T →∞ 1 T Z T 0 u(t)2 dt !1/2 • average-absolute (AA) value: lim T →∞ 1 T Z T 0 |u(t)| dt for some signals these measures can be infinite, or undefined Signals 1–8
- 9. example: 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 the difference, e.g., RMS(u − v) Signals 1–9
- 10. Qualitative 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 t in 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 1–10
- 11. 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: PSfrag replacements t ² 1/² t = 0 PSfrag replacements t 2² 1/² t = 0 • the exact shape of the function doesn’t matter • ² is small (which depends on context) Signals 1–11
- 12. on plots δ is shown as a solid arrow: PSfrag replacements t t = 0 f(t) = δ(t) PSfrag replacements t t = 0 t = −1 f(t) = t + 1 + δ(t) Signals 1–12
- 13. Formal properties formally we define δ by the property that Z b a f(t)δ(t) dt = f(0) provided a < 0, b > 0, and f is continuous at t = 0 idea: 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 • Z b a δ(t) dt = 1 if a < 0 and b > 0 • Z b a δ(t) dt = 0 if a > 0 or b < 0 Signals 1–13
- 14. Z b a δ(t) dt = 0 is ambiguous if a = 0 or b = 0 our convention: to avoid confusion we use limits such as a− or b+ to denote whether we include the impulse or not for example, Z 1 0+ δ(t) dt = 0, Z 1 0− δ(t) dt = 1, Z 0− −1 δ(t) dt = 0, Z 0+ −1 δ(t) dt = 1 Signals 1–14
- 15. Scaled impulses αδ(t − T) is sometimes called an impulse at time T, with magnitude α we have Z b a αδ(t − T)f(t) dt = αf(T) provided a < T < b and f is continuous at T on plots: write magnitude next to the arrow, e.g., for 2δ, PSfrag replacements t 0 2 Signals 1–15
- 16. Sifting property the signal u(t) = δ(t − T) is an impulse function with impulse at t = T for a < T < b, and f continuous at t = T, we have Z b a f(t)δ(t − T) dt = f(T) example: Z 3 −2 f(t)(2 + δ(t + 1) − 3δ(t − 1) + 2δ(t + 3)) dt = 2 Z 3 −2 f(t) dt + Z 3 −2 f(t)δ(t + 1) dt − 3 Z 3 −2 f(t)δ(t − 1) dt + 2 Z 3 −2 f(t)δ(t + 3)) dt = 2 Z 3 −2 f(t) dt + f(−1) − 3f(1) Signals 1–16
- 17. Physical interpretation impulse functions are used to model physical signals • that act over short time intervals • whose effect depends on integral of signal example: 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 • Z 2.001 1.999 f(t) dt = I (mechanical impulse, N · sec) • blow induces change in velocity of v(2.001) − v(1.999) = 1 m Z 2.001 1.999 f(τ) dτ = I/m for (most) applications we can model force as an impulse, at t = 2, with magnitude I Signals 1–17
- 18. example: rapid charging of capacitor PSfrag replacements 1V v(t) i(t) t = 0 1F assuming 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 model Signals 1–18
- 19. for example, include small resistance PSfrag replacements 1V v(t) R v(t) i(t) t = 0 1F i(t) = dv(t) dt = 1 − v(t) R , v(0) = 0 PSfrag replacements R 1 v(t) = 1 − e−t/R PSfrag replacements R 1/R i(t) = e−t/R /R as R → 0, i approaches an impulse, v approaches a unit step Signals 1–19
- 20. as another example, assume the current delivered by the source is limited: if v(t) < 1, the source acts as a current source i(t) = Imax PSfrag replacements Imax v(t) i(t) i(t) = dv(t) dt = Imax, v(0) = 0 PSfrag replacements 1/Imax 1 v(t) PSfrag replacements 1/Imax Imax i(t) as Imax → ∞, i approaches an impulse, v approaches a unit step Signals 1–20
- 21. in conclusion, • large current i acts over very short time between t = 0 and ² • total charge transfer is Z ² 0 i(t) dt = 1 • resulting change in v(t) is v(²) − v(0) = 1 • can approximate i as impulse at t = 0 with magnitude 1 modeling current as impulse • obscures details of current signal • obscures details of voltage change during the rapid charging • preserves total change in charge, voltage • is reasonable model for time scales À ² Signals 1–21
- 22. Integrals of impulsive functions integral of a function with impulses has jump at each impulse, equal to the magnitude of impulse example: u(t) = 1 + δ(t − 1) − 2δ(t − 2); define f(t) = Z t 0 u(τ) dτ PSfrag replacements t u(t) 1 2 t = 1 t = 2 Z t 0 u(τ) dτ f(t) = t for 0 ≤ t < 1, f(t) = t+1 for 1 < t < 2, f(t) = t−1 for t > 2 (f(1) and f(2) are undefined) Signals 1–22
- 23. Derivatives of discontinuous functions conversely, derivative of function with discontinuities has impulse at each jump in function • derivative of unit step function (see page 1–6) is δ(t) • signal f of previous page PSfrag replacements t f(t) 1 2 1 2 3 f0 (t) = 1 + δ(t − 1) − 2δ(t − 2) Signals 1–23
- 24. Derivatives of impulse functions integration by parts suggests we define Z b a δ0 (t)f(t) dt = δ(t)f(t) ¯ ¯ ¯ ¯ b a − Z b a δ(t)f0 (t) dt = −f0 (0) provided a < 0, b > 0, and f0 continuous at t = 0 • δ0 is called doublet • δ0 , δ00 , etc. are called higher-order impulses • similar rules for higher-order impulses: Z b a δ(k) (t)f(t) dt = (−1)k f(k) (0) if f(k) continuous at t = 0 Signals 1–24
- 25. interpretation of doublet δ0 : take two impulses with magnitude ±1/², a distance ² apart, and let ² → 0 PSfrag replacements t = 0 t = ² 1/² 1/² for a < 0, b > 0, Z b a f(t) µ δ(t) ² − δ(t − ²) ² ¶ dt = f(0) − f(²) ² converges to −f0 (0) if ² → 0 Signals 1–25
- 26. Caveat there is in fact no such function (Dirac’s δ is what is called a distribution) • we manipulate impulsive functions as if they were real functions, which they aren’t • it is safe to use impulsive functions in expressions like Z b a f(t)δ(t − T) dt, Z b a f(t)δ0 (t − T) dt provided f (resp, f0 ) is continuous at t = T, and a 6= T, b 6= T • some innocent looking expressions don’t make any sense at all (e.g., δ(t)2 or δ(t2 )) Signals 1–26