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SPFirst-L23

# SPFirst-L23 - Signal Processing First READING ASSIGNMENTS...

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3/27/2004 © 2003, JH McClellan & RW Schafer 1 Signal Processing First Lecture 23 Fourier Transform Properties 3/27/2004 © 2003, JH McClellan & RW Schafer 3 READING ASSIGNMENTS ± This Lecture: ± Chapter 11, Sects. 11-5 to 11-9 ± Tables in Section 11-9 ± Other Reading: ± Recitation: Chapter 11, Sects. 11-1 to 11-9 ± Next Lectures: Chapter 12 (Applications) 3/27/2004 © 2003, JH McClellan & RW Schafer 4 LECTURE OBJECTIVES ± The Fourier transform ± More examples of Fourier transform pairs ± Basic properties of Fourier transforms ± Convolution property ± Multiplication property = dt e t x j X t j ω ) ( ) ( 3/27/2004 © 2003, JH McClellan & RW Schafer 5 Fourier Transform Fourier Analysis ( Forward Transform) = dt e t x j X t j ) ( ) ( Fourier Synthesis ( Inverse Transform) = π d e j X t x t j ) ( 2 1 ) ( ) ( ) ( Domain - Frequency Domain - Time j X t x

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3/27/2004 © 2003, JH McClellan & RW Schafer 6 WHY use the Fourier transform? ± Manipulate the “Frequency Spectrum” ± Analog Communication Systems ± AM: Amplitude Modulation; FM ± What are the “ Building Blocks ” ? ± Abstract Layer , not implementation ± Ideal Filters: mostly BPFs ± Frequency Shifters ± aka Modulators, Mixers or Multipliers: x(t)p(t) 3/27/2004 © 2003, JH McClellan & RW Schafer 7 Frequency Response ± Fourier Transform of h(t) is the Frequency Response ω j j H t u e t h t + = = 1 1 ) ( ) ( ) ( ) ( ) ( t u e t h t = 3/27/2004 © 2003, JH McClellan & RW Schafer 8 2 / ) 2 / sin( ) ( 2 / 0 2 / 1 ) ( T j X T t T t t x = > < = 3/27/2004 © 2003, JH McClellan & RW Schafer 9 > < = = b b b j X t t t x π 0 1 ) ( ) sin( ) (
3/27/2004 © 2003, JH McClellan & RW Schafer 10 0 ) ( ) ( ) ( 0 t j e j X t t t

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SPFirst-L23 - Signal Processing First READING ASSIGNMENTS...

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