Fourier Transform

Fourier Transform - ECE 307 Fourier Transform Z....

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1 ECE 307-2- 1 Fourier Transform Z. Aliyazicioglu Electrical & Computer Engineering Dept. Cal Poly Pomona ECE 307 ECE 307-2- 2 Fourier Transform The Fourier transform (FT) is the extension of the Fourier series to nonperiodic signals. The Fourier transform of a signal exist if satisfies the following condition. The Fourier transform The inverse Fourier transform (IFT) of X( ω ) is x(t)and given by 2 () xt d t −∞ < ∞ ( ) jt X xte d t ω −∞ = 1 ( ) 2 x tX e d π −∞ =
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2 ECE 307-2- 3 Fourier Transform Also, The Fourier transform can be defined in terms of frequency of Hertz as and corresponding inverse Fourier transform is 2 () jf t X fx t e d t π −∞ = 2 t x tX f e d f −∞ = ECE 307-2- 4 Fourier Transform Determine the Fourier transform of a rectangular pulse shown in the following figure Example: -a/2 a/2 h t x(t) /2 22 sin( ) 2 2 sin( ) 2 2 sinc 2 a aa jj jt a h Xh e d t e e j a ha ha a a ha ωω ω ==− == ⎛⎞ = ⎜⎟ ⎝⎠
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3 ECE 307-2- 5 Fourier Transform Example (cont): >> h=1; >> a=1; >> f=-3.5:0.01:3.5; >> w=2*pi*f; >> x=h*a*sinc(w*a/(2*pi)); >> plot (w,x) >> title ('X(\omega)') >> xlabel('\omega'); ω π = = ⎛⎞ = ⎜⎟ ⎝⎠ 1, 1 () 1 s i n c 2 h a X -25 -20 -15 -10 -5 0 5 10 15 20 25 0 0.5 1 |X( ω )| ω -25 -20 -15 -10 -5 0 5 10 15 20 25 0 1 2 3 4 phase X( ω ) ω ECE 307-2- 6 Fourier Transform 2 2 () 2 s i n c 2 h a X = = = >> h=1; >> a=2; >> f=-3.5:0.01:3.5; >> w=2*pi*f; >> x=abs(h*a*sinc(w*a/(2*pi))); >> subplot (2,1,1) >> plot (w,x) >> title ('|X(\omega)|') >> xlabel('\omega') >> xp=phase(h*a*sinc(w*a/(2*pi))); >> subplot (2,1,2) >> plot (w,xp) >> title ('phase X(\omega)') >> xlabel('\omega') -25 -20 -15 -10 -5 0 5 10 15 20 25 0 0.5 1 1.5 2 |X( ω )| ω -25 -20 -15 -10 -5 0 5 10 15 20 25 0 1 2 3 4 phase X( ω ) ω -25 -20 -15 -10 -5 0 5 10 15 20 25 -0.5 0 0.5 1 1.5 2 |X( ω )| ω π - π 2 π -2 π
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4 ECE 307-2- 7 Fourier Transform Example: To find in frequency domain, () /2 22 2 2 sin( ) sin( ) sinc a fa fa jj jf t a h Xf he d t e e hf a fa ha ff a ha fa ππ π == = h=1; a=1; f=-3.5:0.01:3.5; x=h*a*sinc(f*a); plot (f,x) title ('X(f)') xlabel('f'); 1 1 () s i n c h a X = = = -4 -3 -2 -1 0 1 2 3 4 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 X(f) f 1 2 3 -1 -2 -3 ECE 307-2- 8 Fourier Transform Determine the Fourier transform of the Delta function δ (t) Example 0 ( ) 1 jt j Xt e d t e ωω ωδ −− −∞ = 1 X( ω ) ω δ (t) t
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5 ECE 307-2- 9 Fourier Transform Properties of the Fourier Transform We summarize several important properties of the Fourier Transform as follows. 1. Linearity (Superposition ) 11 () ( ) xt X ω 22 X 2 2 1 1 ax t aX +⇔+ Then, If and Proof: [] 1 1 2 2 jt axt axte d t a xte d t ωω ∞∞ −− −∞ −∞ −∞ += + =+ ∫∫ ECE 307-2- 10 Fourier Transform Properties of the Fourier Transform 2. Time Shifting Then, If Proof: X 0 0 ( ) j t xt t X e −⇔ 0 tt τ =− 0 = + dt d = 0 0 0 0 ( ) j xt t e d t x e d ex e d eX ωτ −+ −∞ −∞ −∞ −= = = Let then and
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6 ECE 307-2- 11 Fourier Transform Let 0 () ( ) yt xt t =− 00 0 (( ) ) j tj t jX t YX e X e e Xe ω ωω −− ∠− == = 0 j Xt jY Ye
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This note was uploaded on 03/12/2008 for the course ECE 307 taught by Professor Z.aliyazicioglu during the Winter '07 term at Cal Poly Pomona.

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Fourier Transform - ECE 307 Fourier Transform Z....

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