# Processing and linear systems i win 09 10 pauly 2 1 0

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Unformatted text preview: r N0 &gt; 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 09-10, 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 09-10, 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 discrete-time signals periodic? – Is the sum of periodic continuous-time signals periodic? EE102A:Signal Processing and Linear Systems I; Win 09-10, 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 09-10, Pauly 20 Causal Signals • Causal signals are non-zero only for t ≥ 0 (starts at t = 0, or later) 2 Causal -2 1 -1 0 1 2 t • Noncausal signals are non-zero for some t &lt; 0 (starts before t = 0) Noncausal -2 -1 1 0 1 2 t EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 21 • Anticausal signals are non-zero only for t ≤ 0 (goes backward in time from t = 0) 2 Anticausal -2 -1 EE102A:Signal Processing and Linear Systems I; Win 09-10, 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 09-10, 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 in-phase and quadrature components of z . √ • j = −1 (engineering notation) √ • i = −1 (physics, chemistry, mathematics) EE102A:Signal Processing and Linear Systems I; Win 09-10, 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 09-10, 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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