Isyarat Oleh Risanuri Hidayat Isyarat adalah Isyarat Bernilai real, skalar Fungsi dari variabel waktu Nilai suatu isyarat pada waktu t harus real Contoh isyarat: Tegangan atau arus listrik dalam suatu rangkaian Suara ECG, EEG dll 0/09/52 risanuri 2
Continuous and Discrete Signals A signal x(t) is a continuous-time signal if t is a continuous variable. If t is a discrete variable, that is, x(t) is defined at discrete times, then x(t) is a discrete-time signal. denoted by {x n }or x[n], where n = integer. 0/09/52 risanuri 3 Analog and Digital Signals: If a continuous-time signal x(t) can take on any value in the continuous interval (a, b), where a may be -oo and b may be +oo, then the continuous-time signal x(t) is called an analog signal. If a discrete-time signal x[n] can take on only a finite number of distinct values, then we call this signal a digital signal. 0/09/52 risanuri 4 2
Real and Complex Signals A signal x(t) is a real signal if its value is a real number, and a signal x(t) is a complex signal if its value is a complex number. A general complex signal x( t ) is a function of the 0/09/52 risanuri 5 Deterministic and Random Signals Deterministic signals are those signals whose values are completely specified for any given time. Thus, a deterministic signal can be modeled by a known function of time t. Random signals are those signals that take random values at any given time and must be characterized statistically. 0/09/52 risanuri 6 3
Even and Odd Signals A signal x ( t ) or x[n] is referred to as an even signal if x ( - t ) = x ( t) x [ - n ] = x [ n ] A signal x ( t ) or x[n] is referred to as an odd signal if x ( - t ) = -x( t ) x [ - n ] = -x[ n ] 0/09/52 risanuri 7 Periodic and Non-periodic Signals: A continuous-time signal x ( t ) is said to be periodic with period T if there is a positive nonzero value of T for which Any continuous-time signal which is not periodic is called a non-periodic (or a-periodic ) signal. 0/09/52 risanuri 8 4
Isyarat Periodis Isyarat Periodis Sangat berguna dalam penentuan tanggapan steady state suatu sistem Analog Mempunyai energi tak berhingga, tetapi Daya-nya berhingga (Average Power), P x( t + T ) P = T T = x( t ) 2 X () t dt untuk setiap t 0/09/52 risanuri 9 Isyarat tak Periodis Hanya terjadi sekali Sangat berguna untuk analisis tanggapan sistem digital Mempunyai Energi, E E () t = x dt 0/09/52 risanuri 0 2 5
Energy and Power Signals Consider v(t) to be the voltage across a resistor R producing a current i(t). The instantaneous power p( t) per ohm is defined as Total energy E and average power P on a per-ohm basis are 0/09/52 risanuri Energy and Power Signals For an arbitrary continuous-time signal x(t), the normalized energy content E of x(t) is defined as The normalized average power P of x(t) is defined as 0/09/52 risanuri 2 6
Energy and Power Signals x(t) (or x[n]) is said to be an energy signal (or sequence) if and only if 0 < E < oo, and so P = 0. For example, a signal having only one square pulse is energy signal. A signal that decays exponentially has finite energy, so, it is also an energy signal. The power of an energy signal is 0, because of dividing finite energy by infinite time (or length). x(t) (or x[n]) is said to be a power signal (or sequence) if and only if 0 < P < oo, thus implying that E = oo. For example, sine wave in infinite length is power signal. Signals that satisfy neither property are referred to as neither energy signals nor power signals. 0/09/52 risanuri 3 Unit step Macam-macam Isyarat Dasar U(t)=, t>=0 =0, t<0 u[n]=, n>=0 =0, n<0 Fungsi Kotak x(t)=,-0.5<t<0.5 =0, yg lain 0/09/52 risanuri 4 7
Macam-macam Isyarat Dasar Isyarat ramp r(t)=t, t>=0 =0, yg lain t r ( t) = u( p) dp Isyarat pulse δ(n)=, n=0 =0, yg lain 0/09/52 risanuri 5 Perubahan Isyarat Isyarat asli X(t) -0.5t pergeseran X(t-3) -0.5t penskalaan - 2 refleksi 2 6 X(2t) -t X(-t) -0.5t -0.5-2 0/09/52 risanuri 6 8
Perubahan Isyarat X[n]=(-0.6) n u[n] X[-n] X[n+3] 0/09/52 risanuri 7 Impulse / Fungsi delta δ(t) Impulse f ( t) δ ( t) dt = f (0) Nilai fungsi sangat besar untuk t=0 δ ( t) = 0, t 0 durasi sangat-sangat singkat ε Luas sama dengan (satu) δ ( p) dp =, untuk ε > 0 real ε 0/09/52 risanuri 8 9
Sifat-sifat Fungsi Delta Sifat sampling Sifat shifting x( t) x( a) x( t) x( a) Sifat penskalaan b δ ( at + b) δ ( t + a) a Sifat konvolusi x( t)* = x( u) δ ( t u a) du = x( t a) 0/09/52 risanuri 9 Sifat sampling x( t) x( a) x(t) x(a) a 0/09/52 risanuri 20 0
Sifat shifting x( t) x( a) δ ( t a) x(t) x(a) a 0/09/52 risanuri 2 b δ ( at + b) δ ( t + a) a Sifat penskalaan δ ( at + b) δ (t) b δ ( t + a) a δ (t) b/a b/a 0/09/52 risanuri 22
x( t)* = x( u) δ ( t u a) du = x( t a) Sifat konvolusi δ ( a t) x(t) x( t a) 0/09/52 risanuri 23 Derivative fungsi Delta δ (t) ' δ ( t) ' δ ( t) 0/09/52 risanuri 24 2
Derivative fungsi Delta m δ (t) '' δ ( t) δ (3) ( t) 0/09/52 risanuri 25 x( t ) Isyarat Diskontinyu x( t + ) x( t ) x( t + ) a Isyarat piecewise-kontinyu, isyarat kontinyu kecuali di titiktitik tertentu t i, i=,2,.. Contoh: isyarat kotak, dll 0/09/52 risanuri 26 3
References S. Haykin, B.V Veen, Signals and Systems, John Wiley & Sons, New York, 999 R Hidayat, Isyarat, http://te.ugm.ac.id/~risanuri/ isyaratsystem/index.html Oppenheim, Signals and Systems,. Hwei P Hsu, Signals and Systems, Schaum s Outline Series, McGraw-Hill, New York, 995 0/09/52 risanuri 27 4