2023/2/5
Signals and Systems
Instructor: Chia-Wen Lin (林嘉文)
Email: cwlin@ee.nthu.edu.tw
Course website: https://eeclass.nthu.edu.tw/course/12470
Chapter 4
The Continuous-Time
Fourier Transform
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Continuous
Periodic
Non-periodic
Fourier
Family
Periodic
Discrete
Non-periodic
Fourier Series (Chap. 3)
Fourier Transform (Chap. 4)
Discrete-time Fourier
Series (Discrete Fourier
Transform) (Sec. 3-6)
Discrete-Time Fourier
Transform (Chap. 5)
Chapter 4 The Continuous-Time Fourier
Transform
Sec. 4.0 Introduction
Sec. 4.1 Representation of Aperiodic Signals: The ContinuousTime Fourier Transform
Sec. 4.2 The Fourier Transform for Periodic Signals
Sec. 4.3 Properties of the Continuous-Time Fourier Transform
Sec. 4.4 The Convolution Property
Sec. 4.5 The Multiplication Property
Sec. 4.6 Tables of Fourier Properties and of Basic Fourier
Transform Pairs
Sec. 4.7 Systems Characterized by Linear Constant Coefficient
Differential Equations
Sec. 4.8 Summary
Basic
Properties
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Sec. 4.1 Representation of Aperiodic Signals: The
Continuous-Time Fourier Transform
Key Concepts:
(i) definition of the continuous-time Fourier transform;
(ii) the Fourier series converges to the continuous-time Fourier transform when
𝑇 → ∞;
(iii) the Dirichlet conditions where the continuous-time Fourier transform exists
(absolutely integrable, a finite number of maxima and minima, and a finite
number of discontinuities);
(iv) the sinc function and its relation to the rectangular function
P.285
4.1.1 Development of the Fourier Transform Representation of an
Aperiodic Signal
Fourier series can analyze a periodic signal.
How do we analyze a nonperiodic signal?
P.287
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+∞
𝑥(𝑡)
= 𝑎𝑘 𝑒
1 𝑇/2
−𝑗𝑘𝜔0 𝑡
𝑎𝑘 = න 𝑥(𝑡)𝑒
𝑑𝑡
𝑇 −𝑇/2
where 𝜔0 = 2𝜋/𝑇
𝑗𝑘𝜔0 𝑡
𝑘=−∞
Since 𝑥(𝑡)
= 𝑥(𝑡) for |𝑡| < 𝑇/2, and also, since 𝑥(𝑡) = 0 outside this interval,
𝑎𝑘 =
1 𝑇/2
1 +∞
න
𝑥(𝑡)𝑒 −𝑗𝑘𝜔0 𝑡 𝑑𝑡 = න 𝑥(𝑡)𝑒 −𝑗𝑘𝜔0 𝑡 𝑑𝑡
𝑇 −𝑇/2
𝑇 −∞
Define
+∞
𝑋(𝑗𝜔) = න
−∞
then
𝑎𝑘 =
P.286-287
𝑥(𝑡)𝑒 −𝑗𝜔𝑡 𝑑𝑡
1
𝑋(𝑗𝑘𝜔0 )
𝑇
𝑇 = 4𝑇1
𝑇 = 8𝑇1
𝑇 = 16𝑇1
P.286-287
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Definition 4.1 The Continuous-Time Fourier Transform and Its Inverse
+∞
𝑥(𝑡)𝑒 −𝑗𝜔𝑡 𝑑𝑡
𝑋(𝑗𝜔) = න
Fourier transform:
−∞
Inverse Fourier transform: 𝑥(𝑡) =
1 +∞
න 𝑋(𝑗𝜔)𝑒 𝑗𝜔𝑡 𝑑𝜔
2𝜋 −∞
These two equations are referred to as the Fourier transform pair.
The Fourier transform can be viewed as the Fourier series in the
case where the period 𝑇 → ∞
(the period is infinite)
P.288
Supplement Alternative Definitions of Fourier Transform Pairs
(1) Fourier transform:
𝑥(𝑡) =
Inverse Fourier transform:
1 +∞
න 𝑋(𝑗𝜔)𝑒 −𝑗𝜔𝑡 𝑑𝜔
2𝜋 −∞
+∞
𝑥(𝑡)𝑒 𝑗𝜔𝑡 𝑑𝑡
𝑋(𝑗𝜔) = න
−∞
(2) Fourier transform:
1 +∞
න 𝑋(𝑗𝜔)𝑒 𝑗𝜔𝑡 𝑑𝜔
2𝜋 −∞
𝑥(𝑡) =
1 +∞
න 𝑥(𝑡)𝑒 −𝑗𝜔𝑡 𝑑𝑡
2𝜋 −∞
Inverse Fourier transform: 𝑋(𝑗𝜔) =
(3) Fourier transform:
+∞
𝑥(𝑡) = න
𝑋(𝑗𝑓)𝑒 𝑗2𝜋𝑓𝑡 𝑑𝑓
−∞
Inverse Fourier transform:
+∞
𝑋(𝑗𝑓) = න
𝑥(𝑡)𝑒 −𝑗2𝜋𝑓𝑡 𝑑𝑡
−∞
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4.1.2 Convergence of Fourier Transforms
Dirichlet conditions
(the constraints for the convergence of Fourier transforms)
1. 𝑥(𝑡) be absolutely integrable
+∞
න
𝑥(𝑡) 𝑑𝑡 < ∞
−∞
2. 𝑥(𝑡) have a finite number of maxima and minima within any finite interval.
3. 𝑥(𝑡) have a finite number of discontinuities within any finite interval.
Furthermore, each of these discontinuities must be finite.
P.289
4.1.3 Examples of Continuous-Time Fourier Transforms
[Example 4.1]
𝑥(𝑡) = 𝑒 −𝑎𝑡 𝑢(𝑡)
𝑎>0
∞
∞
𝑋(𝑗𝜔) = න 𝑒
−𝑎𝑡 −𝑗𝜔𝑡
0
𝑋(𝑗𝜔) =
1
,
𝑎 + 𝑗𝜔
𝑒
1
𝑑𝑡 = −
𝑒 −(𝑎+𝑗𝜔)𝑡 ቤ
𝑎 + 𝑗𝜔
0
𝑎>0
P.290
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𝑋(𝑗𝜔) =
1
,
𝑎 + 𝑗𝜔
𝑎>0
P.291
[Example 4.3]
𝑥(𝑡) = 𝛿(𝑡)
∞
𝑋(𝑗𝜔) = න 𝛿(𝑡)𝑒 −𝑗𝜔𝑡 𝑑𝑡 = 1
−∞
Note: Here, we apply the sifting property of the unit impulse function
𝑏
න 𝑦(𝑡)𝛿(𝑡 − 𝑡0 )𝑑𝑡 = 𝑦 𝑡0
if 𝑎 < 𝑡0 < 𝑏
𝑎
P.292
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[Example 4.4]
𝑥(𝑡) = ൝
1,
|𝑡| < 𝑇1
0,
𝑇1 > |𝑡|
𝑇1
𝑋(𝑗𝜔) = න 𝑒 −𝑗𝜔𝑡 𝑑𝑡 = 2
−𝑇1
sin 𝜔 𝑇1
𝜔
Fourier
transform
P.293
[Example 4.5]
𝑋(𝑗𝜔) = ൝
𝑥(𝑡) =
1,
|𝜔| < 𝑊
0,
|𝜔| > 𝑊
1 𝑊 𝑗𝜔𝑡
sin 𝑊 𝑡
න 𝑒
𝑑𝜔 =
2𝜋 −𝑊
𝜋𝑡
Fourier
transform
P.294-295
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Fourier transform for different 𝑊
Fourier
transform
Fourier
transform
Fourier
transform
P.296
Definition 4.2 Sinc Function
sinc(𝜃) =
Specially,
sinc(0) = 1
sin 𝜋 𝜃
𝜋𝜃
sinc(𝑘) = 0
where 𝑘 is a nonzero integer.
P.295
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∞
න 𝑒
−∞
−𝑗𝜔𝑡
∞
න
1 ∞ 𝑗𝜔𝑡
𝑡
න 𝑒
sin 𝑐
𝑑𝑡 = Π 𝜔
2𝜋 −∞
2𝜋
𝜔
Π 𝑡 𝑑𝑡 = sinc
,
2𝜋
𝑒 −𝑗𝜔𝑡 sinc
−∞
𝑡
𝑑𝑡 = 2𝜋Π 𝜔
2𝜋
∞
න 𝑒 −𝑗𝜔𝑡 sinc 𝑡 𝑑𝑡 = Π
−∞
𝜔
2𝜋
where Π(𝑡) means the rectangular function
Π(𝑡) = 1 for |𝑡| < 1/2, Π(𝑡) = 0 otherwise.
P.295-296
Sec. 4.2 The Fourier Transform for Periodic Signals
Key concepts:
(i) The Fourier transform for a periodic signal is equivalent to the
Fourier series. The output is a linear combination of unit
impulse functions.
periodic function
𝔉
impulse train
(ii) If 𝑥(𝑡) is a linear combination of 𝛿(𝑡 − 𝑘𝑇) where 𝑘 is some
integer, then the Fourier transform of 𝑥(𝑡) periodic.
impulse train
𝔉
periodic function
P.296-297
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If 𝑥(𝑡) is periodic
+∞
𝑥(𝑡) = 𝑎𝑘 𝑒 𝑗𝑘𝜔0 𝑡
𝑘=−∞
then
+∞
𝑋(𝑗𝜔) = 2𝜋𝑎𝑘 𝛿(𝜔 − 𝑘𝜔0 )
𝑘=−∞
P.297
[Example 4.6]
Fourier series coefficients
sin 𝑘 𝜔0 𝑇1
𝑎𝑘 =
𝜋𝑘
+∞
𝑋(𝑗𝜔) =
𝑘=−∞
2 sin 𝑘 𝜔0 𝑇1
𝛿(𝜔 − 𝜔0 )
𝑘
P.297-298
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[Example 4.7]
𝑥(𝑡) = sin𝜔0 𝑡
𝑎1 =
1
2𝑗
𝑎−1 = −
1
2𝑗
𝑎𝑘 = 0, 𝑘 ≠ 1 or − 1
1
2
𝑎𝑘 = 0, 𝑘 ≠ 1 or − 1
𝑥(𝑡) = cos𝜔0 𝑡
𝑎1 =
1
2
𝑎−1 =
P.298-299
[Example 4.8]
+∞
𝑥(𝑡) = 𝛿(𝑡 − 𝑘𝑇)
𝑘=−∞
𝑎𝑘 =
1 +𝑇/2
1
න
𝛿(𝑡)𝑒 −𝑗𝑘𝜔0 𝑡 𝑑𝑡 =
𝑇 −𝑇/2
𝑇
(sifting property in Table 2.1)
+∞
2𝜋
2𝜋𝑘
𝑋(𝑗𝜔) =
𝛿 𝜔−
𝑇
𝑇
𝑘=−∞
Fourier
transform
P.299-300
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Sec. 4.3 Properties of the CTFT
Key concepts
Learning several properties of the continuous-time Fourier transform,
including (i) linearity, (ii) time shifting, (iii) frequency shifting, (iv)
conjugate symmetry, (v) symmetry property for a real input, (vi) symmetry
property for a real and even(or odd) input, (vii) differentiation in time, (viii)
differentiation in frequency, (ix) integration, (x) scaling, (xi) time reversal,
and (xii) duality, and (xiii) Parseval’s relation.
These properties will be summarized in Sec. 4.6. Note that the timeshifting and the frequency-shifting properties form a duality pair. The
differentiation in time and the differentiation in frequency properties also
form a duality pair.
P.300
+∞
𝑥(𝑡) 𝑒 −𝑗𝜔𝑡 𝑑𝑡
𝑋(𝑗𝜔) = න
𝑥(𝑡) =
−∞
𝑥(𝑡)
𝔉
1 +∞
න 𝑋(𝑗𝜔) 𝑒 𝑗𝜔𝑡 𝑑𝜔
2𝜋 −∞
𝑋(𝑗𝜔)
4.3.1 Linearity
If
𝑥(𝑡)
𝔉
𝑦(𝑡)
𝑋(𝑗𝜔)
𝔉
𝑌(𝑗𝜔)
then
𝑎𝑥(𝑡) + 𝑏𝑦(𝑡)
𝔉
𝑎𝑋(𝑗𝜔) + 𝑏𝑌(𝑗𝜔)
P.301
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4.3.2 Time Shifting
𝑥(𝑡 − 𝑡0 )
𝔉
𝑒 −𝑗𝜔𝑡0 𝑋(𝑗𝜔)
(Proof):
1 ∞
𝑥(𝑡) =
න 𝑋(𝑗𝜔) 𝑒 𝑗𝜔𝑡 𝑑𝜔
2𝜋 −∞
1 +∞
𝑥(𝑡 − 𝑡0 ) =
න 𝑋(𝑗𝜔)𝑒 𝑗𝜔(𝑡−𝑡0 ) 𝑑𝜔
2𝜋 −∞
1 +∞ −𝑗𝜔𝑡
0 𝑋(𝑗𝜔) 𝑒 𝑗𝜔𝑡 𝑑𝜔
=
න
𝑒
2𝜋 −∞
P.301-302
[Example 4.9]
𝑥(𝑡) =
1
𝑥 (𝑡 − 2.5) + 𝑥2 (𝑡 − 2.5)
2 1
From [Example 4.4]
𝑋1 (𝑗𝜔) =
2 sin( 𝜔/2)
𝜔
𝑋(𝑗𝜔) = 𝑒 −𝑗5𝜔/2
𝑋2 (𝑗𝜔) =
2 sin( 3𝜔/2)
𝜔
sin( 𝜔/2) + 2 sin( 3𝜔/2)
𝜔
P.302-303
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4.3.3 Conjugation and Conjugate Symmetry
If
𝔉
𝑥(𝑡)
𝑋(𝑗𝜔)
then
𝔉
𝑥 ∗ (𝑡)
𝑋 ∗ (−𝑗𝜔)
(Proof):
∗
+∞
∗
𝑋 (𝑗𝜔) = න
𝑥(𝑡)𝑒
−𝑗𝜔𝑡
𝑑𝑡
+∞
+∞
=න
−∞
𝑥 ∗ (𝑡)𝑒 𝑗𝜔𝑡 𝑑𝑡
𝑋 ∗ (−𝑗𝜔) = න
𝑥 ∗ (𝑡)𝑒 −𝑗𝜔𝑡 𝑑𝑡
−∞
−∞
Moreover, if 𝑥(𝑡) is real, then
𝑋(−𝑗𝜔) = 𝑋 ∗ (𝑗𝜔)
[𝑥 𝑡 : real]
P.303-304
If 𝑥(𝑡) is real
𝑋(−𝑗𝜔) = 𝑋 ∗ (𝑗𝜔)
[𝑥 𝑡 : real]
ℛ𝑒{𝑋(𝑗𝜔)} = Re{𝑋(−𝑗𝜔)}
Even{𝑥(𝑡)}
𝔉
Re{𝑋(𝑗𝜔)}
ℐ𝑚{𝑋(𝑗𝜔)} = −Im{𝑋(−𝑗𝜔)}
Odd{𝑥(𝑡)}
𝔉
𝑗Im{𝑋(𝑗𝜔)}
Examples: In Example 4.1,
𝑥(𝑡) = 𝑒 −𝑎𝑡 𝑢(𝑡)
𝑋(𝑗𝜔) =
1
𝑎 + 𝑗𝜔
𝑋(−𝑗𝜔) =
1
= 𝑋 ∗ (𝑗𝜔)
𝑎 − 𝑗𝜔
P.304
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[Example 4.10]
𝑥 𝑡 = 𝑒 −𝑎 𝑡 = 𝑒 −𝑎𝑡 𝑢(𝑡) + 𝑒 𝑎𝑡 𝑢(−𝑡)
From Example 4.1,
𝑒 −𝑎𝑡 𝑢 𝑡
𝔉
1
𝑎 + 𝑗𝜔
If
𝑒 −𝑎𝑡 𝑢(𝑡) + 𝑒 𝑎𝑡 𝑢(−𝑡)
2
−𝑎𝑡
𝑎𝑡
−𝑎|𝑡|
= 𝑒 𝑢(𝑡) + 𝑒 𝑢(−𝑡) = 𝑒
𝑥(𝑡) = 2 Ev 𝑒 −𝑎𝑡 𝑢(𝑡) = 2
𝑋(𝑗𝜔) = 2Re
1
𝑎 − 𝑗𝜔
2𝑎
= 2Re 2
= 2
2
𝑎 + 𝑗𝜔
𝑎 +𝜔
𝑎 + 𝜔2
P.305-306
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4.3.4 Differentiation and Integration
Differentiation
Property
Proof:
𝑑𝑥(𝑡)
𝑑𝑡
𝔉
𝑗𝜔𝑋(𝑗𝜔)
𝑑𝑥(𝑡)
1 +∞
=
න 𝑗𝜔𝑋(𝑗𝜔)𝑒 𝑗𝜔𝑡 𝑑𝜔
𝑑𝑡
2𝜋 −∞
Integration
Property
𝑡
න
𝑥(𝜏)𝑑𝜏
−∞
𝔉
1
𝑋(𝑗𝜔) + 𝜋𝑋(0)𝛿(𝜔)
𝑗𝜔
P.306
[Example 4.11]
𝑔(𝑡) = 𝛿(𝑡)
𝔉
𝐺(𝑗𝜔) = 1
𝑡
𝑥(𝑡) = න 𝑔(𝜏)𝑑𝜏 = 𝑢(𝑡) (unit step function)
−∞
𝑋(𝑗𝜔) =
𝐺(𝑗𝜔)
1
+ 𝜋𝐺(0)𝛿(𝜔) =
+ 𝜋𝛿(𝜔)
𝑗𝜔
𝑗𝜔
P.307
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[Example 4.12]
𝐺(𝑗𝜔) =
2 sin 𝜔
− 𝑒 𝑗𝜔 − 𝑒 −𝑗𝜔
𝜔
𝑋(𝑗𝜔) =
𝐺(𝑗𝜔)
+ 𝜋𝐺(0)𝛿(𝜔)
𝑗𝜔
𝑋(𝑗𝜔) =
2 sin 𝜔 2 cos 𝜔
−
𝑗𝜔 2
𝑗𝜔
since 𝐺(0) = 0
P.307-308
4.3.5 Time and Frequency Scaling
If
𝑥(𝑡)
then since
𝔉
𝑋(𝑗𝜔)
𝑥(𝑡) =
+∞
𝔉{𝑥(𝑎𝑡)} = න
after substituting 𝑎𝑡 by 𝜏
1 +∞
න 𝑋(𝑗𝜔)𝑒 𝑗𝜔𝑡 𝑑𝜔
2𝜋 −∞
𝑥(𝑎𝑡)𝑒 −𝑗𝜔𝑡 𝑑𝑡
−∞
1 +∞
න 𝑥(𝜏)𝑒 −𝑗(𝜔/𝑎)𝜏 𝑑𝜏, 𝑎 > 0
𝑎 −∞
𝔉{𝑥(𝑎𝑡)} =
1 +∞
− න 𝑥(𝜏)𝑒 −𝑗(𝜔/𝑎)𝜏 𝑑𝜏, 𝑎 < 0
𝑎 −∞
𝑥(𝑎𝑡)
P.308-309
𝔉
1
𝑗𝜔
𝑋
|𝑎|
𝑎
Specially, when 𝑎 = −1,
𝑥(−𝑡)
𝔉
𝑋(−𝑗𝜔)
(time reversal property)
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4.3.6 Duality
(1) Duality for Transform Pairs
In summary, if
𝔉
𝑥 𝑡
𝑋 𝑗𝜔
𝑋 𝑗𝑡
then
1
𝑋 𝑗𝑡
2𝜋
𝔉
𝔉
2𝜋𝑥 −𝜔
𝑥 −𝜔
P.309-310
[Example 4.13]
From Example 4.2
Therefore,
(Proof):
2
1 + 𝜔2
𝑒 −|𝑡|
𝔉
𝑒
−|𝑡|
2
= 2𝜋𝑒 −|𝜔|
1 + 𝑡2
1 ∞
2
=
න
𝑒 𝑗𝜔𝑡 𝑑𝜔
2𝜋 −∞ 1 + 𝜔 2
∞
2𝜋𝑒 −|𝑡| = න
−∞
∞
2𝜋𝑒
P.310-311
−|𝜔|
=න
−∞
2
𝑒 −𝑗𝜔𝑡 𝑑𝜔
1 + 𝜔2
2
𝑒 −𝑗𝜔𝑡 𝑑𝑡
1 + 𝑡2
19
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(2) Duality for Properties
𝑑𝑥(𝑡)
𝑑𝑡
𝑥(𝑡 − 𝑡0 )
𝑡
න
𝑥(𝜏)𝑑𝜏
𝔉
−∞
𝔉
𝔉
Duality
𝑗𝜔𝑋(𝑗𝜔)
−𝑗𝑡𝑥(𝑡)
𝑒 −𝑗𝜔𝑡0 𝑋(𝑗𝜔)
𝑒 𝑗𝜔0 𝑡 𝑥(𝑡)
1
𝑋(𝑗𝜔) + 𝜋𝑋(0)𝛿(𝜔)
𝑗𝜔
−
𝔉
𝔉
𝑑𝑋(𝑗𝜔)
𝑑𝜔
𝑋(𝑗(𝜔 − 𝜔0 ))
1
𝑥(𝑡) + 𝜋𝑥(0)𝛿(𝑡)
𝑗𝑡
𝔉
𝜔
න 𝑋(𝑗𝜂)𝑑𝜂
−∞
P.311
4.3.7 Parseval’s Relation
Parseval’s Relation (Energy Preservation)
+∞
|𝑥(𝑡)|2 𝑑𝑡 =
න
−∞
(Proof)
+∞
න
1 +∞
න |𝑋(𝑗𝜔)|2 𝑑𝜔
2𝜋 −∞
+∞
𝑥(𝑡) 2 𝑑𝑡 = න
−∞
𝑥(𝑡)𝑥 ∗ (𝑡)𝑑𝑡
−∞
+∞
=න
−∞
𝑥(𝑡)
1 +∞ ∗
න 𝑋 (𝑗𝜔)𝑒 −𝑗𝜔𝑡 𝑑𝜔 𝑑𝑡.
2𝜋 −∞
+∞
1 +∞ ∗
1 +∞
−𝑗𝜔𝑡
=
න 𝑋 (𝑗𝜔) න 𝑥(𝑡)𝑒
𝑑𝑡 𝑑𝜔 =
න |𝑋(𝑗𝜔)|2 𝑑𝜔
2𝜋 −∞
2𝜋
−∞
−∞
P.312
20
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[Example 4.14]
If 𝑋(𝑗𝜔) is
∞
evaluate (i) 𝐸 = න
𝑥(𝑡) 2 𝑑𝑡 and (ii) 𝐷 =
−∞
∞
𝑥(𝑡) 2 𝑑𝑡 =
𝐸=න
−∞
𝑑
𝑥(𝑡)|𝑡 = 0
𝑑𝑡
1 ∞
5
න 𝑋(𝑗𝜔) 2 𝑑𝜔 =
2𝜋 −∞
8
𝔉
𝑑
𝑥(𝑡)
𝑗𝜔𝑋(𝑗𝜔) = 𝐺(𝑗𝜔)
𝑑𝑡
1 ∞
1 ∞
𝐷 = 𝑔(0) =
න 𝐺(𝑗𝜔)𝑑𝜔 =
න 𝑗𝜔𝑋(𝑗𝜔)𝑑𝜔 = 0
2𝜋 −∞
2𝜋 −∞
𝑔(𝑡) =
P.312-313
If 𝑋(𝑗𝜔) is
evaluate (i) 𝐸 = න
∞
−∞
∞
𝐸=න
−∞
𝑥(𝑡) 2 𝑑𝑡 and (ii) 𝐷 =
𝑑
𝑥(𝑡)|𝑡 = 0
𝑑𝑡
1 ∞
𝑥(𝑡) 𝑑𝑡 =
න 𝑋(𝑗𝜔) 2 𝑑𝜔 = 1
2𝜋 −∞
2
𝔉
𝑑
𝑥(𝑡)
𝑗𝜔𝑋(𝑗𝜔) = 𝐺(𝑗𝜔)
𝑑𝑡
1 ∞
1 ∞
−1
𝐷 = 𝑔(0) =
න 𝐺(𝑗𝜔)𝑑𝜔 =
න 𝑗𝜔𝑋(𝑗𝜔)𝑑𝜔 =
2𝜋 −∞
2𝜋 −∞
2 𝜋
𝑔(𝑡) =
P.313
21
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Sec. 4.4 The Convolution Property
Key concepts:
(i) The convolution property;
(ii) the application of the convolution property for
determining the output of an LTI system;
(iii) convolution with an impulse function is equal to an
identity or a delay operation
P.314
Convolution Property
𝔉
𝑦(𝑡) = ℎ(𝑡) ∗ 𝑥(𝑡)
(Proof):
𝑌(𝑗𝜔) = 𝐻(𝑗𝜔)𝑋(𝑗𝜔)
+∞
If
then
𝑦(𝑡) = න
𝑥(𝜏)ℎ(𝑡 − 𝜏)𝑑𝜏
−∞
+∞
+∞
𝑌(𝑗𝜔) = න
න
−∞
𝑥(𝜏)ℎ(𝑡 − 𝜏)𝑑𝜏 𝑒 −𝑗𝜔𝑡 𝑑𝑡.
−∞
+∞
𝑌(𝑗𝜔) = න
+∞
𝑥(𝜏) න
−∞
ℎ(𝑡 − 𝜏)𝑒 −𝑗𝜔𝑡 𝑑𝑡 𝑑𝜏
−∞
+∞
𝑌(𝑗𝜔) = න
+∞
𝑥(𝜏)𝑒 −𝑗𝜔𝑡 𝐻(𝑗𝜔)𝑑𝜏 = 𝐻(𝑗𝜔) න
−∞
𝑥(𝜏)𝑒 −𝑗𝜔𝑡 𝑑𝜏
−∞
= 𝑋(𝑗𝜔)𝐻(𝑗𝜔)
P.314-315
22
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Three equivalent LTI systems. Here, each block represents an LTI
system with the indicated frequency response.
P.316
4.4.1 Examples
[Example 4.15]
Consider a continuous-time LTI system with impulse response
ℎ(𝑡) = 𝛿(𝑡 − 𝑡0 )
𝐻(𝑗𝜔) = 𝑒 −𝑗𝜔𝑡0
𝑌(𝑗𝜔) = 𝐻(𝑗𝜔)𝑋(𝑗𝜔) = 𝑒 −𝑗𝜔𝑡0 𝑋(𝑗𝜔).
𝑦(𝑡) = 𝑥(𝑡 − 𝑡0 )
In other words,
𝛿(𝑡 − 𝑡0 ) ∗ 𝑥(𝑡) = 𝑥(𝑡 − 𝑡0 )
P.317
23
2023/2/5
[Example 4.16]
𝑦(𝑡) =
𝑑𝑥(𝑡)
𝑑𝑡
𝑌(𝑗𝜔) = 𝑗𝜔𝑋(𝑗𝜔)
The frequency response of a differentiator is
𝐻(𝑗𝜔) = 𝑗𝜔
P.317
[Example 4.18]
ideal lowpass filter
𝐻(𝑗𝜔) = ൝
1 |𝜔| < 𝜔𝑐
0 |𝜔| > 𝜔𝑐
ℎ(𝑡) =
sin𝜔𝑐 𝑡
𝜋𝑡
P.318-319
24
2023/2/5
[Example 4.19]
𝑥(𝑡) = 𝑒 −𝑏𝑡 𝑢(𝑡),
𝑋(𝑗𝜔) =
If
𝑏>0
1
𝑏 + 𝑗𝜔
ℎ(𝑡) = 𝑒 −𝑎𝑡 𝑢(𝑡),
𝐻(𝑗𝜔) =
𝑎>0
1
𝑎 + 𝑗𝜔
𝑦(𝑡) = ℎ(𝑡) ∗ 𝑥(𝑡)
then
𝑌(𝑗𝜔) =
1
(𝑎 + 𝑗𝜔)(𝑏 + 𝑗𝜔)
𝑦(𝑡) =
𝑌(𝑗𝜔) =
1
1
1
−
𝑏 − 𝑎 𝑎 + 𝑗𝜔 𝑏 + 𝑗𝜔
1
𝑒 −𝑎𝑡 𝑢(𝑡) −𝑒 −𝑏𝑡 𝑢(𝑡)
𝑏−𝑎
P.320-321
[Example 4.20]
𝑥(𝑡) =
𝑋(𝑗𝜔) = ቊ
If
then
P.321-322
sin𝜔𝑐 𝑡
𝜋𝑡
sin𝜔𝑖 𝑡
𝜋𝑡
ℎ(𝑡) =
1,
𝜔 ≤ 𝜔𝑖
0, elsewhere
𝐻(𝑗𝜔) = ቊ
1,
0,
𝜔 ≤ 𝜔𝑐
elsewhere
𝑦(𝑡) = ℎ(𝑡) ∗ 𝑥(𝑡)
𝑌(𝑗𝜔) = ቊ
1,
0,
𝜔 ≤ min 𝜔𝑖 , 𝜔𝑐
elsewhere
sin𝜔𝑐 𝑡
,
𝜋𝑡
𝑦 𝑡 =
sin𝜔𝑖 𝑡
,
𝜋𝑡
if 𝜔𝑐 ≤ 𝜔𝑖
if 𝜔𝑖 ≤ 𝜔𝑐
25
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Sec. 4.5 The Multiplication Property
Key concept:
The multiplication property. It and the convolution property form a
duality pair.
1 +∞
𝑟(𝑡) = 𝑠(𝑡)𝑝(𝑡) ↔ 𝑅(𝑗𝜔) =
න 𝑆(𝑗𝜃)𝑃(𝑗(𝜔 − 𝜃))𝑑𝜃
2𝜋 −∞
P.322
𝑟(𝑡) = 𝑠(𝑡)𝑝(𝑡) ↔ 𝑅(𝑗𝜔) =
[Example 4.21]
𝑝(𝑡) = cos𝜔0 𝑡
𝑅(𝑗𝜔) =
1 +∞
න 𝑆(𝑗𝜃)𝑃(𝑗(𝜔 − 𝜃))𝑑𝜃
2𝜋 −∞
𝑃(𝑗𝜔) = 𝜋𝛿(𝜔 − 𝜔0 ) + 𝜋𝛿(𝜔 + 𝜔0 )
1 +∞
1
1
න 𝑆(𝑗𝜃)𝑃(𝑗(𝜔 − 𝜃))𝑑𝜃 = 𝑆(𝑗(𝜔 − 𝜔0 )) + 𝑆(𝑗(𝜔 + 𝜔0 )),
2𝜋 −∞
2
2
P.323-324
26
2023/2/5
[Example 4.21]
1 +∞
𝑟(𝑡) = 𝑠(𝑡)𝑝(𝑡) ↔ 𝑅(𝑗𝜔) =
න 𝑆(𝑗𝜃)𝑃(𝑗(𝜔 − 𝜃))𝑑𝜃
2𝜋 −∞
𝑝(𝑡) = cos𝜔0 𝑡
𝑃(𝑗𝜔) = 𝜋𝛿(𝜔 − 𝜔0 ) + 𝜋𝛿(𝜔 + 𝜔0 )
1 +∞
1
1
𝑅(𝑗𝜔) =
න 𝑆(𝑗𝜃)𝑃(𝑗(𝜔 − 𝜃))𝑑𝜃 = 𝑆(𝑗(𝜔 − 𝜔0 )) + 𝑆(𝑗(𝜔 + 𝜔0 )),
2𝜋 −∞
2
2
P.323-324
[Example 4.22]
𝑔(𝑡) = 𝑟(𝑡)𝑝(𝑡)
𝑟(𝑡) is the output of Example 4.21
𝑝(𝑡) = cos𝜔0 𝑡
+∞
𝐺(𝑗𝜔) =
1
න 𝑅(𝑗𝜃)𝑃(𝑗(𝜔 − 𝜃))𝑑𝜃
2𝜋 −∞
P.324-325
27
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[Example 4.23]
𝑥(𝑡) =
𝑥(𝑡) = 𝜋
sin( 𝑡)
𝜋𝑡
sin( 𝑡) sin( 𝑡/2)
𝜋𝑡 2
sin( 𝑡/2)
𝜋𝑡
1 sin( 𝑡)
sin( 𝑡/2)
𝑋(𝑗𝜔) = 𝔉
∗𝔉
2
𝜋𝑡
𝜋𝑡
P.325
4.5.1 Frequency-Selective Filtering with Variable Center
Frequency
Implementation of a bandpass filter using amplitude
modulation with a complex exponential carrier.
P.326
28
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equivalent bandpass filter
P.326-327
Sec. 4.6 Tables of Fourier Properties and of Basic
Fourier Transform Pairs
Key concept:
The summaries in Tables 4.1 and 4.2.
P.328
29
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(followed by the next page)
P.328
general form of Parseval’s relation:
+∞
න
−∞
𝑥(𝑡)𝑦 ∗ (𝑡)𝑑𝑡 =
1 +∞
න 𝑋(𝑗𝜔)𝑌 ∗ (𝑡)𝑑𝜔
2𝜋 −∞
P.359
30
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P.329
(followed by the next page)
P.329
31
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Sec. 4.7
Systems Characterized by Linear
Constant Coefficient Differential Equations
Key concept:
Using the differentiation property of the Fourier transform to analyze
the linear differential equation with constant coefficients
P.330
linear constant-coefficient differential equation
𝑁
𝑀
𝑘=0
𝑘=0
𝑀
𝑑𝑘 𝑦(𝑡)
𝑑 𝑘 𝑥(𝑡)
𝑎𝑘
=
𝑏
𝑘
𝑑𝑡𝑘
𝑑𝑡𝑘
𝑁
𝑎𝑘 (𝑗𝜔)𝑘 𝑌(𝑗𝜔) = 𝑏𝑘 (𝑗𝜔)𝑘 𝑋(𝑗𝜔)
𝑘=0
𝑘=0
frequency response 𝐻(𝑗𝜔) for an LTI system
𝐻(𝑗𝜔) =
𝑘
𝑌(𝑗𝜔) σ𝑀
𝑘=0 𝑏𝑘 (𝑗𝜔)
= 𝑁
𝑋(𝑗𝜔) σ𝑘=0 𝑎𝑘 (𝑗𝜔)𝑘
P.331-332
32
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[Example 4.25]
𝑑 2 𝑦(𝑡)
𝑑𝑦(𝑡)
𝑑𝑥(𝑡)
+
4
+
3𝑦(𝑡)
=
+ 2𝑥(𝑡)
𝑑𝑡 2
𝑑𝑡
𝑑𝑡
𝐻(𝑗𝜔) =
(𝑗𝜔) + 2
+ 4(𝑗𝜔) + 3
(𝑗𝜔)2
1
1
𝑗𝜔 + 2
2
𝐻(𝑗𝜔) =
=
+ 2
(𝑗𝜔 + 1)(𝑗𝜔 + 3) 𝑗𝜔 + 1 𝑗𝜔 + 3
1
1
ℎ(𝑡) = 𝑒 −𝑡 𝑢(𝑡) + 𝑒 −3𝑡 𝑢(𝑡)
2
2
impulse response
P.331-332
[Example 4.26]
𝑑2 𝑦(𝑡)
𝑑𝑦(𝑡)
𝑑𝑥(𝑡)
+
4
+
3𝑦(𝑡)
=
+ 2𝑥(𝑡)
𝑑𝑡 2
𝑑𝑡
𝑑𝑡
𝐻(𝑗𝜔) =
input
(𝑗𝜔) + 2
+ 4(𝑗𝜔) + 3
(𝑗𝜔)2
𝑥(𝑡) = 𝑒 −𝑡 𝑢(𝑡)
𝑌(𝑗𝜔) = 𝐻(𝑗𝜔)𝑋(𝑗𝜔) =
𝑌(𝑗𝜔) =
𝑗𝜔 + 2
1
𝑗𝜔 + 2
=
.
(𝑗𝜔 + 1)(𝑗𝜔 + 3) 𝑗𝜔 + 1
(𝑗𝜔 + 1)2 (𝑗𝜔 + 3)
𝐴11
𝐴12
𝐴21
+
+
2
𝑗𝜔 + 1 (𝑗𝜔 + 1)
𝑗𝜔 + 3
𝑦(𝑡) =
P.332-333
1
1
ℎ(𝑡) = 𝑒 −𝑡 𝑢(𝑡) + 𝑒 −3𝑡 𝑢(𝑡)
2
2
1
𝐴11 = ,
4
1
𝐴12 = ,
2
𝐴11 = −
1
4
1 −𝑡 1 −𝑡 1 −3𝑡
𝑒 + 𝑡𝑒 − 𝑒
𝑢(𝑡)
4
2
4
33
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Sec. 4.8 Summary
The Fourier transform possesses a wide variety of important
properties that describe how different characteristics of signals are
reflected in their transforms.
Fourier analysis are particularly well suited to the examination of LTI
systems characterized by linear constant-coefficient differential
equations.
P.333
34
