Isyarat Oleh Risanuri Hidayat
Isyarat • Isyarat adalah – 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 01/09/52
risanuri
2
1
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 {xn}or x[n], where n = integer.
01/09/52
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. 01/09/52
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
01/09/52
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. 01/09/52
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]
01/09/52
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.
01/09/52
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 ) = x (t ) P =
1 T
∫
untuk setiap t
X (t ) dt 2
T
01/09/52
risanuri
9
Isyarat tak Periodis • Hanya terjadi sekali • Sangat berguna untuk analisis tanggapan sistem digital • Mempunyai Energi, E
∞
E=
∫ x(t )
2
dt
−∞ 01/09/52
risanuri
10
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
01/09/52
risanuri
11
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
01/09/52
risanuri
12
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. 01/09/52
risanuri
13
Macam-macam Isyarat Dasar • Unit step U(t)=1, t>=0 =0, t<0
u[n]=1, n>=0 =0, n<0 1
1
∞
−∞
∞
−∞
• Fungsi Kotak x(t)=1,-0.5
−∞
01/09/52
1
∞
risanuri
∞
−∞
14
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)=1, n=0 =0, yg lain 1
01/09/52
∞
−∞
risanuri
15
Perubahan Isyarat • pergeseran
• Isyarat asli
X(t) 1
−∞
X(t-3) 1
1-0.5t
-1
∞
2
−∞
X(2t) 1
−∞
01/09/52
-0.5
6
∞
X(-t) 1
1-t 1
2
• refleksi
• penskalaan
1-0.5t
∞
−∞ -2
risanuri
1
1-0.5t
∞
16
8
Perubahan Isyarat • X[-n]
• X[n]=(-0.6)nu[n]
∞
−∞
∞
−∞
• X[n+3]
∞
−∞
01/09/52
risanuri
17
Impulse • Impulse / Fungsi delta δ(t) ∞
∫
∆
f (t )δ (t )dt = f (0)
• Nilai fungsi sangat besar untuk t=0 • durasi sangat-sangat singkat • Luas sama dengan 1 (satu)
01/09/52
∞
−∞
−∞
δ (t ) = 0, t ≠ 0 ε
∫ δ ( p)dp = 1,
untuk ε > 0 real
−ε
risanuri
18
9
Sifat-sifat Fungsi Delta • Sifat penskalaan
• Sifat sampling
δ (at + b) ≡
∞
∫ x(t )δ (t − a) ≡ x(a)
−∞
• Sifat shifting
1 δ (t + ba ) a
• Sifat konvolusi ∆
x(t ) * δ (t − a ) =
x(t )δ (t − a ) ≡ x(a )δ (t − a )
∞
∫ x(u )δ (t − u − a)du
−∞
= x(t − a ) 01/09/52
risanuri
19
Sifat sampling ∞
∫ x(t )δ (t − a ) ≡ x(a )
δ (t − a )
−∞
x(t )
x(a) a
01/09/52
risanuri
20
10
Sifat shifting x (t )δ (t − a ) ≡ x ( a )δ (t − a )
δ (t − a )
x(t )
x(a) a
01/09/52
risanuri
21
Sifat penskalaan δ (at + b) ≡
1 δ (t + ba ) a
δ (at + b) δ (t )
1 a
b/a
01/09/52
δ (t + ba )
δ (t )
b/a
risanuri
22
11
Sifat konvolusi δ (a − t )
∆
x(t ) * δ (t − a ) =
x(t )
∞
∫ x(u )δ (t − u − a)du
−∞
= x(t − a )
x(t − a)
01/09/52
risanuri
23
Derivative fungsi Delta δ ' (t )
δ (t )
−∞
∞ −∞
∞ δ (t ) '
−∞ 01/09/52
risanuri
∞ 24
12
Derivative fungsi Delta δ m (t )
−∞
δ '' ( t )
∞ −∞
δ
(3)
∞
∞
−∞ 01/09/52
(t )
risanuri
25
Isyarat Diskontinyu
x(t − ) ≠ x(t + ) x(t − ) x(t + ) a
Isyarat piecewise-kontinyu, isyarat kontinyu kecuali di titiktitik tertentu ti, i=1,2,.. Contoh: isyarat kotak , dll
01/09/52
risanuri
26
13
References • S. Haykin, B.V Veen, Signals and Systems, John Wiley & Sons, New York, 1999 • 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, 1995 01/09/52
risanuri
27
14
