# T yt dne jn0t ht h n jn

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Unformatted text preview: ) −∞ = ∞ ￿ ￿ ∞ ￿ n=−∞ jnω0 t Dne = ￿ ∞ dτ h(τ )e−jnω0τ dτ −∞ n=−∞ ∞ ￿ Dnejnω0(t−τ ) ￿ (DnH (jnω0)) ejnω0t n=−∞ Same frequencies as input, diﬀerent complex multipliers. EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 9 Frequency Domain Representation of Signals Since complex exponentials are so easy to analyze for LTI systems, we’d like to represent arbitrary signals as linear combinations of complex exponentials. We studied several diﬀerent cases: • If x(t) is periodic, or of ﬁnite duration, the natural representation is the Fourier series x(t) = ∞ ￿ Dnejnω0t n=−∞ Dn = 1 T0 ￿ τ +T0 x(t)e−jnω0tdt τ where T0 is the fundamental period, and ω0 = EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 2π T0 . 10 • If x(t) is aperiodic, but is an energy or power signal, then the natural representation is the Fourier transform ￿∞ 1 x(t) = X (j ω )ej ωtdω 2π − ￿∞ X (j ω ) = x(t)e−j ωtdt −∞ • If x(t) is not a power signal, we use the Laplace transform ￿ c+j ∞ 1 x(t) = X (s)est ds 2π j c−j ∞ ￿∞ X (s) = x(t)e−stdt −∞ EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 11 Although the speciﬁcs diﬀer, all of these have similar characteristics • Expanding a signal in time, compresses it in frequency • Shifting a signal in the time domain is multiplication by a complex exponential in the transform domain • Multiplying a signal by a complex exponential in the time domain shifts the transform in the frequency domain • Convolution in the time domain becomes multiplication in the frequency domain • Multiplication in the time domain becomes convolution in the frequency domain EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 12 Convergence The Fourier representation converge to the signal in the sense that the mean square error goes to zero. For example, if we just consider the ﬁrst ±N terms of the Fourier series, then the mean square error is EN = ￿ t0 +T0 t0 ˆ |fN (t) − f (t)|2 dt The truncated Fourier series converges to the signal if the integral square error EN → 0 as N → ∞. This does not mean that this relation necessarily holds for all t, only in the integral average sense is it true. EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 13 Example Truncated Fourier Series approximation to a square wave: 2 Terms 1.2 1 0.8 0.6 0.4 0.2 0 !0.2 !2 !1.5 !1 !0.5 0 0.5 1 1.5 2 4 Terms 1.2 1 0.8 0.6 0.4 0.2 0 !0.2 !2 !1.5 !1 !0.5 0 EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 0.5 1 1.5 2 14 8 Terms 1.2 1 0.8 0.6 0.4 0.2 0 !0.2 !2 !1.5 !1 !0.5 0 0.5 1 1.5 2 16 Terms 1.2 1 0.8 0.6 0.4 0.2 0 !0.2 !2 !1.5 !1 !0.5 0 0.5 1 EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 1.5 2 15 Frequency domain representations • Converge to the midpoint of a discontinuity • Oscillate at either side of discontinuity (Gibbs eﬀect) • Increasing num...
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