Overview of Continuous-Time Fourier Transform Topics Definition Compare & contrast with Laplace transform Conditions for existence Relationship to LTI systems Examples Ideal lowpass filters Relationship to DTFS Core Concepts Review x(t) H(s) y(t) x(t) h(t) Laplace transform enables us to find the transient and steady-state response for arbitrary input signals t>0 Bode plots show us how an LTI system responds in steady-state to a collection of sinusoidal input signals Fourier series enables us to represent periodic signals as a sum of harmonically related sinusoids Fourier transforms enable us to represent (almost any) signal as an infinite sum (integral) of non-harmonically related sinusoids Enables us to think about all signals (periodic and non-periodic) as a sum of sinusoids Since sinusoids are eigenfunctions of LTI systems, this representation makes systems analysis easier and intuitive y(t) J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 1 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 2 Orthogonality Defined Two non-periodic power signals x 1 (t) and x 2 (t) are orthogonal if and only if 1 T lim x 1 (t)x T 2T 2(t)dt =0 Orthogonality of Complex Sinusoids Consider two (possibly non-harmonic) complex sinusoids Are they orthogonal? lim T 1 2T x 1 (t) =e jω 1t x 1 (t)x 1 T 2(t) = lim N 2T = lim = N 1 2T x 2 (t) =e jω 2t { 1 ω 1 = ω 2 0 Otherwise e jω 1t e jω 2t dt e j(ω 1 ω 2 )t dt J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 3 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 4
Importance of Orthogonality Suppose that we know a signal is composed of a linear combination of non-harmonic complex sinusoids x(t) = 1 X(e jω )e jωt dω How do we solve for the coefficients X(e jω )? lim T = lim T = 1 x(t)e jω ot dt [ 1 [ X(e jω ) ] X(e jω )e jωt dω e jωot dt lim T ] e jωt e jωot dt dω Workspace = 1 [ ] T X(e jω ) lim e j(ω ωo)t dt dω T = 1 [ X(e jω e j(ω ωo)t e j(ω ω o)t ) lim T j(ω ω o ) = 1 = 1 = = X(e jω o ) [ X(e jω ) lim 2sin[(ω ω o)t ] T ω ω o X(e jω )δ(ω ω o )dω X(e jω ) δ(ω ω o )dω ] dω ] dω J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 5 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 6 Definition and Comments F{x(t)} = X(jω) F 1 {X(jω)} = x(t) 1 x(t)e jωt dt X(jω)e jωt dω Denote relationship as x(t) FT X(jω) X(jω) can be thought of as a density of x(t) at the frequency ω Used to characterize LTI systems and to analyze signals Some books & engineers define it differently Most ECE texts use the same definition we are using We will use this definition exclusively Mean Squared Error F{x(t)} = X(jω) F 1 {X(jω)} = ˆx(t) 1 MSE = lim T 1 2T +T x(t)e jωt dt X(jω)e jωt dω x(t) ˆx(t) 2 dt Like the Fourier series, it can be shown that if the transform converges, X(jω) minimizes the MSE over all possible functions of ω Like the other transforms, the error converges to zero Like the CTFS, MSE = 0 does not imply x(t) =ˆx(t) for all t The functions may differ at points of discontinuity J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 7 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 8
Example 4: Applying the Definition Find the Fourier transform of x(t) =δ(t). X(jω)= X(s) = Compare & Contrast with Laplace Transform x(t)e jωt dt x(t) = 1 0 x(t)e st dt x(t)u(t) = 1 j X(jω)e jωt dω σ1 +j σ 1 j X(s)e st ds Unlike the Laplace transform, - FT has no mechanism to accommodate initial conditions - FT is more difficult to find the transient response - FT does not converge for as many signals (e.g., e t u(t)) + FT can be applied to two-sided signals (x(t) 0for t 0) + FT is easier for steady-state problems + FT provides more insight for analysis of frequency content J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 9 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 10 Conditions for Existence The Fourier transform of a signal x(t) exists if x(t) 2 dt< and any discontinuities are finite True for all signals of finite amplitude and duration Do periodic signals have a Fourier transform? No, but we can still apply the transform if we allow X(jω) to be expressed in terms of impulse functions This requires the trick of using limits As with CTFS, convergence does not imply that the inverse Fourier transform will recover the signal However, will be equal at all points except for discontinuities Separate sufficient conditions (Dirichlet) are stated in the text Fourier Transform & Transfer Functions M k=0 Y (jω)= b k(jω) k 1+ N k=1 a X(jω)=H(jω) X(jω) k k(jω) The time-domain relationship of y(t) and x(t) can be complicated In the frequency domain, the relationship of Y (jω) to X(jω) of LTI systems described by differential equations simplifies to a rational function of (jω) The numerator/denomenator sums are not Fourier series For real systems, H(jω) is usually a rational ratio of two polynomials H(jω) is the discrete-time transfer function Specifically, the transfer function of an LTI system can be defined as the ratio of Y (jω) to X(jω) Same story as the continuous-time case J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 11 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 12
Example 5: Relationship to CT LTI Systems Suppose the impulse response h(t) is known for an LTI CT system. Derive the relationship between a sinusoidal input signal and the output of the system. Example 5: Workspace J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 13 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 14 Example 6: Transform of a Decaying Exponential Let x(t) =e at u(t) where Re{a} > 0. Does the Fourier transform of x(t) exist? Find the Fourier transform (use a limit, if necessary). Example 6: Workspace J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 15 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 16
1 Example 6: Fourier Transform e 1.0t u(t) function [] = DecayingExponential(); a = 1; w = -20:0.1:20; X = 1./(a + j*w); Example 6: MATLAB Code X(jω) X(jω) 0.5 100 50 0 20 15 10 5 0 5 10 15 20 50 0 FigureSet(1, LTX ); subplot(2,1,1); h = plot(w,abs(x)); set(h, LineWidth,1.5); ylabel( X(j\omega) ); title(sprintf( e^{-%5.1ft} u(t),a)); ylim([0 1.1]); box off; AxisLines; subplot(2,1,2); h = plot(w,angle(x)*180/pi); set(h, LineWidth,1.5); ylabel( \angle X(j\omega) ); xlabel( Frequency (rad/s) ); box off; AxisLines; AxisSet(8); print -depsc DecayingExponential; 100 20 15 10 5 0 5 Frequency (rad/s) 10 15 20 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 17 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 18 Example 7: Transform of a Pulse Does the Fourier transform of p T (t) defined below exist? Find the Fourier transform (use a limit if necessary). { 1 t <T p T (t) = 0 Otherwise Example 7: Workspace J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 19 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 20
Example 7: Pulse Spectrum Example 7: MATLAB Code 2 Pulse Fourier Transform for T = 1 1.5 X(jω) 1 0.5 0 0.5 60 40 20 0 20 40 60 Frequency (rad/s) J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 21 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 22 Example 8: Inverse Transform of a Pulse Solve for the signal x(t), given X(jω) below. { 1 ω <W X(jω)= 0 Otherwise Example 8: Workspace J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 23 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 24
Example 8: Workspace Continued Example 9: Inverse Transform of an Impulse Find the inverse Fourier transform of an impulse X(jω)=δ(ω ω o ). J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 25 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 26 Example 10: Transform of Periodic Signal Find the Fourier transform of a periodic signal x(t) = + k= X[k]ejkω 0t. Example 10: Workspace J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 27 J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 28
Summary of Key Concepts The Fourier transform can be viewed as a special case of the two-sided Laplace transform X(jω)=X(s) s=jω Although it is less general, it is easier and more intuitive to work with for signal processing applications (e.g., communications) Most engineers working in this field are more familiar with the Fourier transform Periodic signals have a CTFT that consists of impulses at multiples of the fundamental frequency J. McNames Portland State University ECE 223 CT Fourier Transform Ver. 1.24 29