Linear Systems lecture 7 Fourier transforms academic year : 16-17 - PowerPoint PPT Presentation

Linear Systems lecture 7 Fourier transforms academic year : 16-17 lecture : 7 UNIVERSITY OF TWENTE ✳ build : November 23, 2016 slides : 31

Today UNIVERSITY OF TWENTE ✳ The Fourier transform The Dirac delta function Properties of the Fourier transform Nicolet TM iS TM 50 Fourier Transform Infra Red Spectrometer Linear Systems The Fourier transform 1 LS.16-17[7] 23-11-2016 2 The fundamental theorem of Fourier transforms 1 intro Properties of the Fourier transform 3 LS

The Fourier transform UNIVERSITY OF TWENTE ✳ The Fourier transform The Dirac delta function x ( t ) x ( t ) x ( t ) Properties of the t Fourier transform The Fourier transform of a time-continuous, non-periodic signal x ( t ) is derived form the Fourier series of an approximating periodic signal. Linear Systems LS.16-17[7] 23-11-2016 2 1.1 LS

The Fourier transform UNIVERSITY OF TWENTE ✳ x ( t ) x ( t ) x ( t ) ˜ ˜ ˜ The Fourier transform The Dirac delta function x ( t ) x ( t ) x ( t ) Properties of the t Fourier transform − T − T T T 2 2 2 2 The Fourier transform of a time-continuous, non-periodic signal x ( t ) is derived form the Fourier series of an approximating periodic signal. For T > 0 define the periodic approximation ˜ x ( t ) by if − T 2 < t < T x ( t ) = x ( t ) ˜ 2 , Linear Systems LS.16-17[7] where the period is T . 23-11-2016 2 1.1 LS

The Fourier transform UNIVERSITY OF TWENTE ✳ x ( t ) x ( t ) x ( t ) ˜ ˜ ˜ The Fourier transform The Dirac delta function x ( t ) x ( t ) x ( t ) Properties of the t Fourier transform − T − T T T 2 2 2 2 The Fourier transform of a time-continuous, non-periodic signal x ( t ) is derived form the Fourier series of an approximating periodic signal. For T > 0 define the periodic approximation ˜ x ( t ) by if − T 2 < t < T x ( t ) = x ( t ) ˜ 2 , Linear Systems LS.16-17[7] where the period is T . 23-11-2016 Let T → ∞ , and analyse what happens with the 2 1.1 Fourier series of ˜ x ( t ) . LS

The Fourier transform UNIVERSITY OF TWENTE ✳ x ( t ) x ( t ) x ( t ) ˜ ˜ ˜ The Fourier transform The Dirac delta function x ( t ) x ( t ) x ( t ) Properties of the t Fourier transform − T − T T T 2 2 2 2 The Fourier transform of a time-continuous, non-periodic signal x ( t ) is derived form the Fourier series of an approximating periodic signal. For T > 0 define the periodic approximation ˜ x ( t ) by if − T 2 < t < T x ( t ) = x ( t ) ˜ 2 , Linear Systems LS.16-17[7] where the period is T . 23-11-2016 Let T → ∞ , and analyse what happens with the 2 1.1 Fourier series of ˜ x ( t ) . LS

From series to integral UNIVERSITY OF TWENTE ✳ Let F be a function defined on the real numbers. Consider the sum The Fourier transform ∞ � The Dirac delta F ( n ∆ ω )∆ ω. function n = −∞ Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 3 1.2 LS

From series to integral UNIVERSITY OF TWENTE ✳ Let F be a function defined on the real numbers. Consider the sum The Fourier transform ∞ � The Dirac delta F ( n ∆ ω )∆ ω. function n = −∞ Properties of the Fourier transform ∆ ω F ω − 2∆ ω − ∆ ω ∆ ω 2∆ ω 3∆ ω 4∆ ω 5∆ ω 6∆ ω 0 Linear Systems By regarding the sum as a Riemann sum, we see LS.16-17[7] 23-11-2016 � ∞ ∞ � 3 1.2 lim F ( n ∆ ω )∆ ω = F ( ω ) d ω. ∆ ω → 0 + −∞ n = −∞ LS

The Fourier integral UNIVERSITY OF TWENTE ✳ Define ∆ ω = 2 π/ T , then from the Fundamental theorem of Fourier series follows: � � � T / 2 ∞ The Fourier 1 � x ( τ ) e − in ∆ ωτ d τ transform e in ∆ ω t x ( t ) = ˜ T The Dirac delta − T / 2 n = −∞ function Properties of the Fourier transform Linear Systems LS.16-17[7] 23-11-2016 4 1.3 LS

The Fourier integral UNIVERSITY OF TWENTE ✳ Define ∆ ω = 2 π/ T , then from the Fundamental theorem of Fourier series follows: � � � T / 2 ∞ The Fourier 1 � x ( τ ) e − in ∆ ωτ d τ transform e in ∆ ω t x ( t ) = ˜ T The Dirac delta − T / 2 n = −∞ function �� T / 2 � ∞ Properties of the = 1 � x ( τ ) e in ∆ ω ( t − τ ) d τ Fourier transform ∆ ω 2 π − T / 2 n = −∞ Linear Systems LS.16-17[7] 23-11-2016 4 1.3 LS

The Fourier integral UNIVERSITY OF TWENTE ✳ Define ∆ ω = 2 π/ T , then from the Fundamental theorem of Fourier series follows: � � � T / 2 ∞ The Fourier 1 � x ( τ ) e − in ∆ ωτ d τ transform e in ∆ ω t x ( t ) = ˜ T The Dirac delta − T / 2 n = −∞ function �� T / 2 � ∞ Properties of the = 1 � x ( τ ) e in ∆ ω ( t − τ ) d τ Fourier transform ∆ ω 2 π − T / 2 n = −∞ � T / 2 � ∞ T → ∞ ≈ �� ∞ ∞ � − T / 2 −∞ x ( t ) = 1 � x ( τ ) e in ∆ ω ( t − τ ) d τ ∆ ω 2 π −∞ n = −∞ � �� � F ( n ∆ ω ) Linear Systems LS.16-17[7] 23-11-2016 4 1.3 LS

The Fourier integral UNIVERSITY OF TWENTE ✳ Define ∆ ω = 2 π/ T , then from the Fundamental theorem of Fourier series follows: � � � T / 2 ∞ The Fourier 1 � x ( τ ) e − in ∆ ωτ d τ transform e in ∆ ω t x ( t ) = ˜ T The Dirac delta − T / 2 n = −∞ function �� T / 2 � ∞ Properties of the = 1 � x ( τ ) e in ∆ ω ( t − τ ) d τ Fourier transform ∆ ω 2 π − T / 2 n = −∞ � T / 2 � ∞ T → ∞ ≈ �� ∞ ∞ � − T / 2 −∞ x ( t ) = 1 � x ( τ ) e in ∆ ω ( t − τ ) d τ ∆ ω 2 π −∞ n = −∞ � �� � F ( n ∆ ω ) � ∞ �� ∞ � = 1 x ( τ ) e i ω ( t − τ ) d τ d ω 2 π −∞ −∞ Linear Systems LS.16-17[7] 23-11-2016 4 1.3 LS

The Fourier integral UNIVERSITY OF TWENTE ✳ Define ∆ ω = 2 π/ T , then from the Fundamental theorem of Fourier series follows: � � � T / 2 ∞ The Fourier 1 � x ( τ ) e − in ∆ ωτ d τ transform e in ∆ ω t x ( t ) = ˜ T The Dirac delta − T / 2 n = −∞ function �� T / 2 � ∞ Properties of the = 1 � x ( τ ) e in ∆ ω ( t − τ ) d τ Fourier transform ∆ ω 2 π − T / 2 n = −∞ � T / 2 � ∞ T → ∞ ≈ �� ∞ ∞ � − T / 2 −∞ x ( t ) = 1 � x ( τ ) e in ∆ ω ( t − τ ) d τ ∆ ω 2 π −∞ n = −∞ � �� � F ( n ∆ ω ) � ∞ �� ∞ � = 1 x ( τ ) e i ω ( t − τ ) d τ d ω 2 π −∞ −∞ Linear Systems � ∞ �� ∞ � LS.16-17[7] = 1 x ( τ ) e − i ωτ d τ e i ω t d ω 23-11-2016 2 π −∞ −∞ 4 1.3 � �� � X ( ω ) LS

The Fourier transform UNIVERSITY OF TWENTE ✳ Definition Let x ( t ) be a continuous-time signal. Then the Fourier The Fourier transform of x ( t ) is defined as transform � ∞ The Dirac delta x ( t ) e − i ω t d t , function X ( ω ) = Properties of the −∞ Fourier transform provided this integral exists. Linear Systems LS.16-17[7] 23-11-2016 5 1.4 LS

The Fourier transform UNIVERSITY OF TWENTE ✳ Definition Let x ( t ) be a continuous-time signal. Then the Fourier The Fourier transform of x ( t ) is defined as transform � ∞ The Dirac delta x ( t ) e − i ω t d t , function X ( ω ) = Properties of the −∞ Fourier transform provided this integral exists. The Fourier transform X ( ω ) is sometimes called the spectrum of x . Linear Systems LS.16-17[7] 23-11-2016 5 1.4 LS

The Fourier transform UNIVERSITY OF TWENTE ✳ Definition Let x ( t ) be a continuous-time signal. Then the Fourier The Fourier transform of x ( t ) is defined as transform � ∞ The Dirac delta x ( t ) e − i ω t d t , function X ( ω ) = Properties of the −∞ Fourier transform provided this integral exists. The Fourier transform X ( ω ) is sometimes called the spectrum of x . Convention: the name of the Fourier transform is the uppercase form of the of the signal. Linear Systems LS.16-17[7] 23-11-2016 5 1.4 LS

The Fourier transform UNIVERSITY OF TWENTE ✳ Definition Let x ( t ) be a continuous-time signal. Then the Fourier The Fourier transform of x ( t ) is defined as transform � ∞ The Dirac delta x ( t ) e − i ω t d t , function X ( ω ) = Properties of the −∞ Fourier transform provided this integral exists. The Fourier transform X ( ω ) is sometimes called the spectrum of x . Convention: the name of the Fourier transform is the uppercase form of the of the signal. Linear Systems Alternative notation: X ( ω ) = F { x ( t ) } . LS.16-17[7] 23-11-2016 5 1.4 LS

The Fourier transform UNIVERSITY OF TWENTE ✳ Definition Let x ( t ) be a continuous-time signal. Then the Fourier The Fourier transform of x ( t ) is defined as transform � ∞ The Dirac delta x ( t ) e − i ω t d t , function X ( ω ) = Properties of the −∞ Fourier transform provided this integral exists. The Fourier transform X ( ω ) is sometimes called the spectrum of x . Convention: the name of the Fourier transform is the uppercase form of the of the signal. Linear Systems Alternative notation: X ( ω ) = F { x ( t ) } . LS.16-17[7] 23-11-2016 If X ( ω ) is the Fourier transform of x ( t ) , then we denote 5 1.4 this as x ( t ) ↔ X ( ω ) . LS