# Dt t 2 1 t t 2 t 2 tej 2ntt dt t

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Fourier series is then ∞ ￿1 f (t) = ej 2πnt/T T n=−∞ = ∞ 1 ￿ jnω0t e . T n=−∞ EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 9 The Fourier transform of δT (t) is then F [δT (t)] = ∞ 1￿ 2πδ (ω − nω0) T n=−∞ = ω0 ∞ ￿ n=−∞ δ (ω − nω0) = ω0δω0 (ω ) since ω0 = 2π /T . We then have the transform pair δT (t) ⇔ ω0δω0 (ω ) The Fourier transform of an array evenly spaced δ ’s is another array of evenly spaced δ ’s! EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 10 The delta train δT (t) is !T (t ) 1 -3T -2T -T 0 T 2T 3T t and its Fourier transform ω0δω0 (t) is !0"!0 (!) !0 −3!0 −2!0 −!0 !0 0 !0 = 2"/T 2!0 3!0 ! These are very useful functions! EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 11 Ideal Sampling One of the most important uses of δT (t) is to represent sampling. If f (t) is a signal, then f (t)δT (t) = f (t) ∞ ￿ n=−∞ = ∞ ￿ n=−∞ = ∞ ￿ n=−∞ δ (t − nT ) f (t)δ (t − nT ) f (nT )δ (t − nT ) where we have used the fact that f (t)δ (t − T ) = f (T )δ (t − T ) for the last step. EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 12 Original Signal -3T f (t ) -2T -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 T EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 13 Example: Square Wave, Revisited We can write the square wave function from earlier today as a convolution: f (t) = rect(t) ∗ δ2(t) which is illustrated below rect(t ) −3 −3 −2 −2 −1 −1 0 ∗ 0 = 1 2 3t 2 3t 2 3t !2(t ) 1 f (t ) −3 −2 −1 0 EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 1 14 Using the convolution theorem, the Fourier transform of f (t) is F [f (t)] = F [rect(t)] F [δ2(t)] = sinc(ω /2π )ω0δω0 (ω ) = π sinc(ω /2π )δπ (ω ) where ω0 = 2π /T = 2π /2 = π . To get this into the form we found earlier, expand δπ (ω ) F [f (t)] = π sinc(ω /2π ) =π ∞ ￿ n=−∞ =π ∞ ￿ n=−∞ ∞ ￿ n=−∞ δ (ω − π n) sinc(ω /2π )δ (ω − π n) sinc(n/2)δ (ω − π n) EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 15 which is the same thing we obtained before. In general, 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 ) Recall that 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 09-10, Pauly 16 This is what we saw the periodic rect signal example: F1( j!) −12!0 −8!0 −4!0 !0F1( j!)"!0 (!) 0 4!0 8!0 EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 12!0 ! 17 Sampling Theorem What is the Fourier transform of a signal that has been sampled...
View Full Document

## 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.

Ask a homework question - tutors are online