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N0 > 0 such that
x[n + N0] = x[n] for all n N0 is the period of x[n] in sample spacings.
• The smallest T0 or N0 is the fundamental period of the periodic signal.
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 17 Example: 2 x(t ) 1
2 1 0 1 1
1 0 t x(t − 1) 2 2 2 1 2 t Shifting x(t) by 1 time unit results in the same signal.
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 18 • Common periodic signals are sines and cosines
x(t) = A cos(2π t/T0 − θ) x[n] = A cos(2π n/N0 − θ)
• An aperiodic signal is a signal that is not periodic.
• Seems like a simple concept, but there are some interesting cases
– Is x[n] = A cos(2π na − θ) periodic for any a?
– Is the sum of periodic discretetime signals periodic?
– Is the sum of periodic continuoustime signals periodic? EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 19 Periodic Extension
• Periodic signals can be generated by periodic extension by any segment
of length one period T0 (or a multiple of the period).
2 One Period x1(t ) 1
2 1 0 1 2 t 2
Periodic Extension x(t )
1 2 1 0 1 2 t • We will often take a signal that is deﬁned only over an interval T0 and
use periodic extension to make a periodic signal.
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 20 Causal Signals
• Causal signals are nonzero only for t ≥ 0 (starts at t = 0, or later)
2
Causal
2 1
1 0 1 2 t • Noncausal signals are nonzero for some t < 0 (starts before t = 0)
Noncausal 2 1 1
0 1 2 t EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 21 • Anticausal signals are nonzero only for t ≤ 0 (goes backward in time
from t = 0)
2
Anticausal 2 1 EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 1
0 1 2 t 22 Complex Signals • So far, we have only considered real (or integer) valued signals.
• Signals can also be complex
z (t) = x(t) + jy (t)
where x(t) and y (t) are each real valued signals, and j = √ −1. • Arises naturally in many problems
– Convenient representation for sinusoids
– Communications
– Radar, sonar, ultrasound EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 23 Review of Complex Numbers Complex number in Cartesian form: z = x + jy
• x = z , the real part of z • y = z , the imaginary part of z
• x and y are also often called the inphase and quadrature components
of z .
√
• j = −1 (engineering notation)
√
• i = −1 (physics, chemistry, mathematics) EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 24 Complex number in polar form: z = rej φ
• r is the modulus or magnitude of z
• φ is the angle or phase of z
• exp(j φ) = cos φ + j sin φ Im z = x + jy Im z = re j! r
!
Re
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly Re
25 • complex exponential of z = x + jy :
ez = ex+jy = exejy = ex(cos y + j sin y )
Know how to add, multiply, and divide complex numbers, and be able to
go between represe...
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 Fall '13
 Mukamel
 Signal Processing

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