S. Boyd EE102 Lecture 1 Signals notation and meaning ² common signals ² size of a signal ² qualitative properties of signals ² impulsive signals ² 1{1 Signals a signal is a function of time, e.g., f is the force on some mass ² v is the output voltage of some circuit ² out p is the acoustic pressure at some point ² notation: f, v , p or f( ), v ( ), p( ) refer to the whole signal or function ² out ¢ out ¢ ¢ f(t), vout(1:2), p(t + 2) refer to the value of the signals at times t, 1:2, ² and t + 2, respectively for times we usually use symbols like t, ¿, t1, . Signals 1{2 Example 1 a) (P 0 ) t ( p PSfrag replacements −1 −1 0 1 2 3 t (msec) Signals 1{3 Domain of a signal domain of a signal: t's for which it is de¯ned some common domains: all t, i.e., R ² nonnegative t: t 0 ² (here t = 0 just means¸ some starting time of interest) t in some interval: a t b ² · · t at uniformly sampled points: t = kh + t , k = 0; 1; 2; : : : ² 0 § § discrete-time signals are de¯ned for integer t, i.e., t = 0; 1; 2; : : : ² (here t means sample time or epoch, not real time in seconds)§ § we'll usually study signals de¯ned on all reals, or for nonnegative reals Signals 1{4 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) ² vector signal: u(t) is a vector of some dimension ² binary signal: u(t) is either 0 or 1 ² we'll usually encounter scalar signals example: a vector-valued signal v1 v = v 2 2 3 v3 4 5 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 unspeci¯ed Signals 1{5 Common signals with names a constant (or static or DC) signal: u(t) = a, where a is some constant ² the unit step signal (sometimes denoted 1(t) or U(t)), ² u(t) = 0 for t < 0; u(t) = 1 for t 0 ¸ the unit ramp signal, ² u(t) = 0 for t < 0; u(t) = t for t 0 ¸ a rectangular pulse signal, ² u(t) = 1 for a t b; u(t) = 0 otherwise · · a sinusoidal signal: ² u(t) = a cos(!t + Á) a, b, !, Á are called signal parameters Signals 1{6 Real signals most real signals, e.g., AM radio signal ² FM radio signal ² cable TV signal ² audio signal ² NTSC video signal ² 10BT ethernet signal ² telephone signal ² aren't given by mathematical formulas, but they do have de¯ning characteristics Signals 1{7 Measuring the size of a signal size of a signal u is measured in many ways for example, if u(t) is de¯ned for t 0: ¸ 1 integral square (or total energy): u(t)2 dt ² Z0 squareroot of total energy ² 1 integral-absolute value: u(t) dt ² j j Z0 peak or maximum absolute value of a signal: max ¸ u(t) ² t 0 j j 1=2 1 T root-mean-square (RMS) value: lim u(t)2 dt ² T !1 T Ã Z0 ! 1 T average-absolute (AA) value: lim u(t) dt ² T !1 T j j Z0 for some signals these measures can be in¯nite, or unde¯ned Signals 1{8 example: for a sinusoid u(t) = a cos(!t + Á) for t 0 ¸ the peak is a ² j j the RMS value is a =p2 0:707 a ² j j ¼ j j the AA value is a 2=¼ 0:636 a ² j j ¼ j j the integral square and integral absolute values are ² 1 the deviation between two signals u and v can be found as the size of the di®erence, e.g., RMS(u v) ¡ Signals 1{9 Qualitative properties of signals u decays if u(t) 0 as t ² ! ! 1 u converges if u(t) a as t (a is some constant) ² ! ! 1 u is bounded if its peak is ¯nite ² u is unbounded or blows up if its peak is in¯nite ² u is periodic if for some T > 0, u(t + T ) = u(t) holds for all t ² in practice we are interested in more speci¯c quantitative questions, e.g., how fast does u decay or converge? ² how large is the peak of u? ² Signals 1{10 Impulsive signals (Dirac's) delta function or impulse ± is an idealization of a signal that is very large near t = 0 ² is very small away from t = 0 ² has integral 1 ² for example: PSfrag replacements ² PSfrag replacements 2² 1=² 1=² t t t = 0 t = 0 the exact shape of the function doesn't matter ² ² is small (which depends on context) ² Signals 1{11 on plots ± is shown as a solid arrow: f(t) = ±(t) PSfrag replacements t t = 0 f(t) = t + 1 + ±(t) PSfrag replacements t t = ¡1 t = 0 Signals 1{12 Formal properties formally we de¯ne ± by the property that b f(t)±(t) dt = f(0) Za provided a < 0, b > 0, and f is continuous at t = 0 idea: ± acts over a time interval very small, over which f(t) f(0) ¼ ±(t) = 0 for t = 0 ² 6 ±(0) isn't really de¯ned ² b ±(t) dt = 1 if a < 0 and b > 0 ² Za b ±(t) dt = 0 if a > 0 or b < 0 ² Za Signals 1{13 b ±(t) dt = 0 is ambiguous if a = 0 or b = 0 Za our convention: to avoid confusion we use limits such as a or b+ to denote whether we include the impulse or not ¡ for example, 1 1 0¡ 0+ ±(t) dt = 0; ±(t) dt = 1; ±(t) dt = 0; ±(t) dt = 1 Z0+ Z0¡ Z¡1 Z¡1 Signals 1{14 Scaled impulses ®±(t T ) is sometimes called an impulse at time T , with magnitude ® ¡ we have b ®±(t T )f(t) dt = ®f(T ) ¡ Za provided a < T < b and f is continuous at T on plots: write magnitude next to the arrow, e.g., for 2±, 2 PSfrag replacements t 0 Signals 1{15 Sifting property the signal u(t) = ±(t T ) is an impulse function with impulse at t = T ¡ for a < T < b, and f continuous at t = T , we have b f(t)±(t T ) dt = f(T ) ¡ Za example: 3 f(t)(2 + ±(t + 1) 3±(t 1) + 2±(t + 3)) dt ¡ ¡ Z¡2 3 3 3 = 2 f(t) dt + f(t)±(t + 1) dt 3 f(t)±(t 1) dt ¡ ¡ Z¡2 Z¡2 Z¡2 3 + 2 f(t)±(t + 3)) dt Z¡2 3 = 2 f(t) dt + f( 1) 3f(1) ¡ ¡ Z¡2 Signals 1{16 Physical interpretation impulse functions are used to model physical signals that act over short time intervals ² whose e®ect depends on integral of signal ² 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 ² 2:001 f(t) dt = I (mechanical impulse, N sec) ² ¢ Z1:999 blow induces change in velocity of ² 1 2:001 v(2:001) v(1:999) = f(¿) d¿ = I=m ¡ m Z1:999 for (most) applications we can model force as an impulse, at t = 2, with magnitude I Signals 1{17 example: rapid charging of capacitor PSfrag replacements t = 0 i(t) 1V v(t) 1F assuming v(0) = 0, what is v(t), i(t) for t > 0? i(t) is very large, for a very short time ² a unit charge is transferred to the capacitor `almost instantaneously' ² v(t) increases to v(t) = 1 `almost instantaneously' ² to calculate i, v, we need a more detailed model Signals 1{18 PSfrag replacements for example, include small resistance v(t) R i(t) 1V v(t) t = 0 1F dv(t) 1 v(t) i(t) = = ¡ ; v(0) = 0 dt R PSfrag replacements1 PSfrag replacements1=R ¡t=R v(t) = 1 ¡ e i(t) = e¡t=R=R R R as R 0, i approaches an impulse, v approaches a unit step ! Signals 1{19 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 i(t) Imax v(t) dv(t) i(t) = = I ; v(0) = 0 dt max v(t) i(t) PSfrag replacements1 PSfrag replacementsImax 1=Imax 1=Imax as I , i approaches an impulse, v approaches a unit step max ! 1 Signals 1{20 in conclusion, large current i acts over very short time between t = 0 and ² ² ² total charge transfer is i(t) dt = 1 ² Z0 resulting change in v(t) is v(²) v(0) = 1 ² ¡ can approximate i as impulse at t = 0 with magnitude 1 ² modeling current as impulse obscures details of current signal ² obscures details of voltage change during the rapid charging ² preserves total change in charge, voltage ² is reasonable model for time scales ² ² À Signals 1{21 Integrals of impulsive functions integral of a function with impulses has jump at each impulse, equal to the magnitude of impulse t PSfragexample:replacementsu(t) = 1 + ±(t 1) 2±(t 2); de¯ne f(t) = u(¿) d¿ ¡ ¡ ¡ Z0 1 u(t) t t t = 1 t = 2 u(¿) d¿ Z0 2 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 unde¯ned) Signals 1{22 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 ±(t) ² signal f of previous page ² f(t) PSfrag replacements 3 2 1 t 1 2 f 0(t) = 1 + ±(t 1) 2±(t 2) ¡ ¡ ¡ Signals 1{23 Derivatives of impulse functions integration by parts suggests we de¯ne b b b ±0(t)f(t) dt = ±(t)f(t) ±(t)f 0(t) dt = f 0(0) ¡ ¡ Za ¯a Za ¯ ¯ provided a < 0, b > 0, and f 0 continuous¯ at t = 0 ±0 is called doublet ² ±0, ±00, etc.