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Unformatted text preview: UCSB Fall 2007 Handout on Fourier Transforms This handout supplements the coverage of Fourier transforms in the text and lectures. 1 Hertz or Radians/second? We wish to be fluent in working with both angular frequency ω in radians/second, and frequency f in Hertz, where ω = 2 πf . Working with ω is often shorter than carrying around 2 πf but working with f makes the duality between the time and frequency domains more transparent. The main thing to watch out for is that dω = 2 πdf , which impacts us when doing frequency domain integrals, such as the inverse Fourier transform, frequency domain convolution (corresponding to multiplication in time), and computing Parseval’s identity. 2 A physical interpretation of the Fourier transform We begin with an important physical interpretation of the Fourier transform of the impulse response of an LTI system.. Complex Exponential through LTI system: If a complex exponential x ( t ) = e jωt is passed through an LTI system with impulse response h ( t ), then the output y ( t ) is given by y ( t ) = ( x * h )( t ) = integraltext ∞-∞ x ( t- τ ) h ( τ ) dτ = integraltext ∞-∞ e jω ( t- τ ) h ( τ ) dτ = e jωt integraltext ∞-∞ h ( τ ) e- jωτ dτ = H ( jω ) e jωt (1) Transfer function: For an LTI system with impulse response h ( t ), the Fourier transform H ( jω ) = H ( j 2 πf ) is called the transfer function. As you have seen in circuits classes, we often plot the magnitude | H ( jω ) | and phase arg ( H ( jω )) as a function of frequency. LTI systems with real-valued impulse response: For an LTI system with real-valued impulse response h ( t ), the transfer function is conjugate symmetric: H (- jω ) = H * ( jω ). In other words, if H ( jω ) = Ge jθ (where G ≥ 0), then H (- jω ) = Ge- jθ . Now, suppose that the input to the system is a real-valued sinusoid of the form x ( t ) = cos( ωt + φ ). x ( t ) = 1 2 ( e j ( ωt + φ ) + e- j ( ωt + φ ) ) = 1 2 e jφ e jωt + A 2 e- jφ e- jωt Using (1) and linearity, we obtain that the system output is given by y ( t ) = 1 2 e jθ H ( jω ) e jωt + 1 2 e- jθ H (- jω ) e- jωt = 1 2 e jφ Ge jθ e jωt + 1 2 e- jφ Ge- jθ e- jωt = G 2 ( e j ( ωt + φ + θ ) + e- j ( ωt + φ + θ ) ) = G cos( ωt + φ + θ ) (2) To summarize, for an LTI system with real-valued impulse response, the response to a sinusoid is a sinusoid at the same frequency, scaled by the magnitude of the transfer function, and its phase is shifted by the phase of the transfer function, evaluated at the frequency of the sinusoid. Figure 1 is a pictorial depiction of equations (1) and (2), showing the physical significance of the transfer function of an LTI system. h(t) real ω H(j ) ω exp(j t) ω LTI system h(t) ω cos( t+ ) ϕ arg(H(j )) ω |H(j )| ω cos( t+ + ) ω ϕ LTI system exp(j t) Figure 1: The physical significance of the transfer function of an LTI system....
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