Representations converge to the midpoint of a

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Unformatted text preview: ber of terms compresses ringing, but doesn’t reduce its amplitude! EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 16 Impulse Trains One of the most important signals we talked about was the impulse train, δT (t) = ∞ ￿ n=−∞ δ (t − nT ) These are self-transforming δT (t) ⇔ ω0δω0 (ω ) The Fourier transform of an array evenly spaced δ ’s is another array of evenly spaced δ ’s! Connects Fourier series and Fourier transform. Models sampling. EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 17 Periodic Sequences If f1(t) is one cycle of a periodic function with period T , then f (t) is f (t) = f1(t) ∗ δT (t) and the Fourier transform is F (j ω ) = ω0F1(j ω )δω0 (ω ) Multiplying by δω0 (ω ) samples F1(j ω ) at multiples of ω0. The Fourier transform of the periodic signal is the sampled Fourier transform of one period EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 18 Square wave, period 2. Convolution of rect(t) and the impulse train δ2(t) rect(t ) −3 −3 −2 −2 −1 −1 0 1 ∗ 2 3t 2 3t 2 3t !2(t ) 0 1 = f (t ) −3 −2 −1 0 1 EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 19 Periodic signal spectrum is sampled spectrum of one cycle F1( j!) −12!0 −8!0 −4!0 !0F1( j!)"!0 (!) 0 4!0 8!0 12!0 ! Periodic signals have sampled spectra. EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 20 Samping and Reconstruction We can represent sampling as multiplying by an impulse train ¯ f (t) = f (t)δT (t) = ∞ ￿ n=−∞ f (nT )δ (t − nT ) In the frequency domain, the corresponds to periodization ∞ ∞ ￿ 1￿ ¯ (j ω ) = 1 F (j ω ) ∗ F δ (ω − nω0) = F (j (ω − nω0)) T T n=−∞ n=−∞ EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 21 Time domain: Original Signal -3T -2T f (t ) -T 0 T 2T 3T t 2T 3T t !T (t ) 1 -2T 3T t × Sampling Function -3T 2T -T 0 T = Sampled Signal -3T -2T f (t )!T (t ) -T 0 EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly T 22 Frequency domain: F ( j!) −!0 −2!B 0 2!B ∗ −!0 ! 1 !" (") T0 0 ! = 1 F ( j(! + !0)) T 1 F ( j!) T −!0 lowpass ﬁlter −2!B 0 ¯ F (!) 1 F ( j(! − !0)) T 2!B ! We can perfectly recover F (j ω ) (and f (t)) if the sampling rate the Nyquist rate. EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 1 T > 2B , 23 As the sampling frequency ω0 decreases (sampling period T increases) the spectral replicas get closer: 2!B −2!B −!0 −2!0 −3!0 −2!0 0 −!0 −!0 0 0 !0 !0 !0 ! 2!0 ! 2!0 3!0 ! Eventually the replicas overlap, and F (j ω ) cannot be recovered. EE102A:Signal Processing and Linear Systems I; Win 08-09 Pauly 24 To recover f (t) we use an ideal lowpass ﬁlter with 2B = 1/T , sampling at the Nyquist rate T rect ￿!￿ 4"B 1 F (!) T −!0 −!0/2 !0 ! !0/2 2!B 0 −2!B The lowpass ﬁlter has the Fourier transform H (j ω ) = T rect ￿ω￿ 4π B which has the impulse response h(t...
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This note was uploaded on 09/28/2013 for the course EE 108b taught by Professor Mukamel during the Fall '13 term at Singapore Stanford Partnership.

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