Topic 5: Discrete-Time Fourier Transform (DTFT) o DT Fourier - PowerPoint PPT Presentation

ELEC361: Signals And Systems Topic 5: Discrete-Time Fourier Transform (DTFT) o DT Fourier Transform o Overview of Fourier methods o DT Fourier Transform of Periodic Signals o Properties of DT Fourier Transform o Relations among Fourier Methods o Summary o Appendix: o Transition from DT Fourier Series to DT Fourier Dr. Aishy Amer Transform Concordia University Electrical and Computer Engineering Figures and examples in these course slides are taken from the following sources: • A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 • M.J. Roberts, Signals and Systems, McGraw Hill, 2004 • J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

DT Fourier Transform � (Note: a Fourier transform is unique, i.e., no two same signals in time give the same function in frequency) � The DT Fourier Series is a good analysis tool for systems with periodic excitation but cannot represent an aperiodic DT signal for all time � The DT Fourier Transform can represent an aperiodic discrete-time signal for all time � Its development follows exactly the same as that of the Fourier transform for continuous-time aperiodic signals 2

DT Fourier Transform Let x[n] be the aperiodic DT signal � We construct a periodic signal ˜x[n] for � which x[n] is one period ˜x[n] is comprised of infinite number of � replicas of x[n] Each replica is centered at an integer � multiple of N N is the period of ˜x[n] � Consider the following figure which � illustrates an example of x[n] and the construction of Clearly, x[n] is defined between − N 1 and � N 2 Consequently, N has to be chosen such � that N > N 1 + N 2 + 1 so that adjacent replicas do not overlap Clearly, as we let � as desired 3

DT Fourier Transform � Let us now examine the FS representation of � Since x[n] is defined between − N 1 and N 2 � a k in the above expression simplifies to ω = 2 π /N 4

DT Fourier Transform � Now defining the function � We can see that the coefficients a k are related to X(e j ω ) as � where ω 0 = 2 π /N is the spacing of the samples in the frequency domain � Therefore � As N increases ω 0 decreases, and as N → ∞ the above equation becomes an integral 5

DT Fourier Transform � One important observation here is that the function X(e j ω ) is periodic in ω with period 2 π � Therefore, as N → ∞ , � (Note: the function e j ω is periodic with N=2 π ) � This leads us to the DT-FT pair of equations 6

DT Fourier Transform: Forms ⊗ ⇒ + DT Fourier Transform : DT CT P π 2 ∞ ∑ ω = − ω j j n X ( e ) x [ n ] e = −∞ n ⊗ + ⇒ Inverse DT Fourier Transform : CT P DT π 2 1 ∫ = ω ω ω j j n x [ n ] X ( e ) e d 7 π 2 2 π

DT Fourier Transform: Examples � 1 � 1 8

DT Fourier Transform: Example 9

Outline o DT Fourier Transform o Overview of Fourier methods o DT Fourier Transform of Periodic Signals o Properties of DT Fourier Transform o Relations among Fourier Methods o DTFT: Summary o Appendix: o Transition from DT Fourier Series to DT Fourier Transform 10

Overview of Fourier Analysis Methods: Types of signals 11

Overview of Fourier Analysis Methods: Types of signals 12

Overview of Fourier Analysis Methods: Continuous-Value and Continuous-Time Signals All continuous signals are � CT but not all CT signals are continuous 13

Overview of Fourier Analysis Methods Periodic in Time Aperiodic in Time Discrete in Frequency Continuous in Frequency ⊗ ⇒ CT Fourier Transform : CT CT Continuous ⊗ ⇒ CT Fourier Series : CT - P DT T ∞ in Time T ∫ − ω ω = 1 j t ∫ X ( j ) x ( t ) e dt − ω = jk t a x ( t ) e dt 0 k T − ∞ 0 ⊗ ⇒ Inverse CT Fourier Transform : CT CT ⊗ ⇒ Aperiodic in CT Inverse Fourier Series : DT CT - P T ∞ Frequency ∞ 1 ∫ ∑ ω = ω ω ω j t = x ( t ) X ( j ) e d jk t x ( t ) a e 0 π k 2 = −∞ − ∞ k ⊗ ⇒ ⊗ ⇒ + Discrete in DT Fourier Series DT - P DT - P DT Fourier Transform : DT CT P π N N 2 Time − ∞ N 1 ∑ ∑ − ω ω − ω = = j kn j j n X [ k ] x [ n ] e X ( e ) x [ n ] e 0 = −∞ = n 0 n ⊗ ⇒ ⊗ + ⇒ Inverse DT Fourier Transform : CT P DT Inverse DT Fourier Series DT - P DT - P π N N 2 − Periodic in N 1 1 ∑ 1 ω = ∫ ω ω j kn = ω x [ n ] X [ k ] e j j n 0 x [ n ] X ( e ) e d Frequency π 2 N 2 = k 0 π 14

Outline o DT Fourier Transform o Overview of Fourier methods o DT Fourier Transform of Periodic Signals o Properties of DT Fourier Transform o Relations among Fourier Methods o DTFT: Summary o Appendix: o Transition from DT Fourier Series to DT Fourier Transform 15

Fourier Transform of Periodic DT Signals � Consider the continuous time signal � This signal is periodic � Furthermore, the Fourier series representation of this signal is just an impulse of weight one centered at ω = ω 0 � Now consider this signal � It is also periodic and there is one impulse per period However, the separation between adjacent impulses is 2 π , which agrees with the properties of DT Fourier Transform � In particular, the DT Fourier Transform for this signal is 16

Fourier Transform of Periodic DT Signals: Example � 1 � The signal can be expressed as � We can immediately write � Or equivalently where X(e j ω ) is periodic in ω with period 2 π 17

Outline o DT Fourier Transform o Overview of Fourier methods o DT Fourier Transform of Periodic Signals o Properties of DT Fourier Transform o Relations among Fourier Methods o DTFT: Summary o Appendix: o Transition from DT Fourier Series to DT Fourier Transform 18

Properties of the DT Fourier Transform � Note: the function e j ω is periodic with N=2 π 19

Properties of the DT Fourier Transform 20

Properties of the DT Fourier Transform 21

Properties of the DT Fourier Transform 22

Properties of the DT Fourier Transform 23

Properties of the DT Fourier Transform 24

Properties of the DT Fourier Transform 25

Properties of the DT Fourier Transform 26

Properties of the DT Fourier Transform 27

Properties of the DT Fourier Transform 28

Properties of the DT Fourier Transform 29

Properties of the DT Fourier Transform 30

Properties of the DT Fourier Transform 31

Properties of the DT Fourier Transform 32

Properties of the DT Fourier Transform: Example 33

Properties of the DT Fourier Transform 34

Properties of the DT Fourier Transform: Difference equation � DT LTI Systems are characterized by Linear Constant- Coefficient Difference Equations � A general linear constant-coefficient difference equation for an LTI system with input x[n] and output y[n] is of the form � Now applying the Fourier transform to both sides of the above equation, we have � But we know that the input and the output are related to each other through the impulse response of the system, denoted by h[n], i.e., 35

Properties of the DT Fourier Transform : Difference equation � Applying the convolution property � if one is given a difference equation corresponding to some system, the Fourier transform of the impulse response of the system can found directly from the difference equation by applying the Fourier transform � Fourier transform of the impulse response = Frequency response � Inverse Fourier transform of the frequency response = Impulse response 36

Properties of the DT Fourier Transform: Example � With |a| < 1 , consider the causal LTI system that us characterized by the difference equation � From the discussion, it is easy to see that the frequency response of the system is � From tables (or by applying inverse Fourier transform), one can easily find that 37

Outline o DT Fourier Transform o Overview of Fourier methods o DT Fourier Transform of Periodic Signals o Properties of DT Fourier Transform o Relations among Fourier Methods o DTFT: Summary o Appendix: o Transition from DT Fourier Series to DT Fourier Transform 38