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Mar 08, 2024 104 likes •205 views

Overview of Continuous-Time Fourier Transform Topics Core Concepts Review Definition H ( s ) h ( t ) x ( t ) y ( t ) x ( t ) y ( t ) Compare & contrast with Laplace transform Laplace transform enables us to find the transient and

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SLIDE 1

Orthogonality Defined Two non-periodic power signals x1(t) and x2(t) are orthogonal if and

  • nly if

lim

T →∞

1 2T T

−T

x1(t)x∗

2(t) dt = 0

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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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
  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Orthogonality of Complex Sinusoids Consider two (possibly non-harmonic) complex sinusoids x1(t) = ejω1t x2(t) = ejω2t Are they orthogonal? lim

T →∞

1 2T T

−T

x1(t)x∗

2(t) = lim N→∞

1 2T T

−T

ejω1te−jω2t dt = lim

N→∞

1 2T T

−T

ej(ω1−ω2)t dt =

  • 1

ω1 = ω2 Otherwise

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Core Concepts Review H(s)

x(t) y(t)

h(t)

x(t) y(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

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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SLIDE 2

Definition and Comments F {x(t)} = X(jω) +∞

−∞

x(t) e−jωt dt F−1 {X(jω)} = x(t) 1 2π +∞

−∞

X(jω) ejωt 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
  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Importance of Orthogonality Suppose that we know a signal is composed of a linear combination of non-harmonic complex sinusoids x(t) = 1 2π ∞

−∞

X(ejω) ejωt dω How do we solve for the coefficients X(ejω)? lim

T →∞

T

−T

x(t)e−jωot dt = lim

T →∞

T

−T

1 2π ∞

−∞

X(ejω) ejωt dω

  • e−jωot dt

= 1 2π ∞

−∞

X(ejω)

  • lim

T →∞

T

−T

ejωte−jωot dt

  • dω
  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Mean Squared Error F {x(t)} = X(jω) +∞

−∞

x(t) e−jωt dt F−1 {X(jω)} = ˆ x(t) 1 2π +∞

−∞

X(jω) ejωt dω MSE = lim

T →∞

1 2T +T

−T

|x(t) − ˆ x(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

  • f ω
  • Like the other transforms, the error converges to zero
  • Like the CTFS, MSE = 0 does not imply x(t) = ˆ

x(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

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Workspace = 1 2π ∞

−∞

X(ejω)

  • lim

T →∞

T

−T

ej(ω−ωo)t dt

  • dω

= 1 2π ∞

−∞

X(ejω)

  • lim

T →∞

ej(ω−ωo)T − e−j(ω−ωo)T j(ω − ωo)

  • dω

= 1 2π ∞

−∞

X(ejω)

  • lim

T →∞ 2sin[(ω − ωo)T]

ω − ωo

  • dω

= 1 2π ∞

−∞

X(ejω) 2π δ(ω − ωo) dω = ∞

−∞

X(ejω) δ(ω − ωo) dω = X(ejωo)

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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SLIDE 3

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
  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 4: Applying the Definition Find the Fourier transform of x(t) = δ(t).

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Fourier Transform & Transfer Functions Y (jω) = M

k=0 bk(jω)k

1 + N

k=1 ak(jω)k X(jω) = H(jω) X(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

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Compare & Contrast with Laplace Transform X(jω) = +∞

−∞

x(t)e−jωt dt x(t) = 1 2π ∞

−∞

X(jω)ejωt dω X(s) = ∞

0− x(t)e−st dt

x(t)u(t) = 1 2πj σ1+j∞

σ1−j∞

X(s)est 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., etu(t))

+ FT can be applied to two-sided signals (x(t) = 0 for 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

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SLIDE 4

Example 6: Transform of a Decaying Exponential Let x(t) = e−atu(t) where Re{a} > 0. Does the Fourier transform of x(t) exist? Find the Fourier transform (use a limit, if necessary).

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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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

  • utput of the system.
  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 6: Workspace

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 5: Workspace

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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SLIDE 5

Example 7: Transform of a Pulse Does the Fourier transform of pT (t) defined below exist? Find the Fourier transform (use a limit if necessary). pT (t) =

  • 1

|t| < T Otherwise

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 6: Fourier Transform

−20 −15 −10 −5 5 10 15 20 0.5 1 |X(jω)| e− 1.0t u(t) −20 −15 −10 −5 5 10 15 20 −100 −50 50 100 ∠ X(jω) Frequency (rad/s)

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 7: Workspace

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 6: MATLAB Code

function [] = DecayingExponential(); a = 1; w = -20:0.1:20; X = 1./(a + j*w); 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;

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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SLIDE 6

Example 8: Inverse Transform of a Pulse Solve for the signal x(t), given X(jω) below. X(jω) =

  • 1

|ω| < W Otherwise

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 7: Pulse Spectrum

−60 −40 −20 20 40 60 −0.5 0.5 1 1.5 2 Frequency (rad/s) X(jω) Pulse Fourier Transform for T = 1

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 8: Workspace

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 7: MATLAB Code

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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SLIDE 7

Example 10: Transform of Periodic Signal Find the Fourier transform of a periodic signal x(t) = +∞

k=−∞ X[k]ejkω0t.

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 8: Workspace Continued

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

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Example 10: Workspace

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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Example 9: Inverse Transform of an Impulse Find the inverse Fourier transform of an impulse X(jω) = δ(ω − ωo).

  • J. McNames

Portland State University ECE 223 CT Fourier Transform

  • Ver. 1.24

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SLIDE 8

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

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