Definition of Discrete-Time Fourier Transform (DTFT)

Transcription

1 Definition of Discrete-Time ourier Transform (DTT) {x[n]} = X(e jω ) + n= {X(e jω )} = x[n] x[n]e jωn Why use the above awkward notation for the transform? X(e jω )e jωn dω Answer: It is consistent with the z-transform X(z) x[n]z n X(e jω ) = X(z) z=e jω n= This also enables us to distinguish between the discrete-time (DT) and continuous-time (CT) ourier transforms. Dr. H. Nguyen Page 23

2 X(e jω ) = n= } {{ } Analysis equation Some Observations x[n]e jωn ; x[n] = X(e jω )e jωn dω } {{ } Synthesis equation Since e jωn = e j(ω+l)n, for any pair of integers l and n X(e jω ) is a periodic function of ω with a fundamental period of. Unlike the DT ourier series, the frequency ω is continuous (Recall that the CT ourier transform X(jω) is not periodic in general). Thus the DT synthesis integral can be taken over any continuous interval of length This is similar to the CT ourier series analysis equation Dr. H. Nguyen Page 24

3 requency Concept for Discrete-Time Signals Recall that e j(ω+l)n = e jωn for any integer l If seemingly very high-frequency discrete-time signals, cos[(ω + l)n], are equal to low-frequency discrete-time signals, cos(ωn), what do low and high frequencies mean for discrete-time signals? Note that the unit of ω is radians per sample A sinusoid with a frequency of. radians per sample is the same as one with a frequency of. + radian per sample No DT signal can oscillate faster between two consecutive samples of opposite magnitudes No DT signal can oscillate slower than radians per sample Thus (i) ω = π, l(π + ) is highest perceivable frequency for DT signals (ii) ω =, l() is lowest perceivable frequency Dr. H. Nguyen Page 25

4 5 EE35 Spectrum Analysis and Discrete Time Systems Portland State University ECE 223 Discrete-Time ourier Transform Ver..4 6 requency Concept for Discrete-Time Signals (Cont d) Discrete-Time requency Concepts Continued X(e jω ) 4π 3π π π 3π 4π ω X(e jω ) 4π 3π π π 3π 4π ω Low frequencies are those that are near Low frequencies are those that are near High frequencies are those near ±π High frequencies are those near ±π Intermediate frequencies are those in between Intermediate frequencies are those in between Note that that the the highest highest frequency, frequency, π radians π per radians sampleper is equal sample to.5iscycles equal per to.5 cycles per sample sample We will encounter this concept again when we discuss sampling 7 Dr. H. Nguyen Page 26 Portland State University ECE 223 Discrete-Time ourier Transform Ver..4 8

5 Examples Example : ind the ourier transform of h[n] = a n u[n] where a <. Sketch the transform over range of 3π to 3π for a =.5 and a =.5. Example 2: ind the ourier transform of the following pulse signal. Sketch the transform over range of 3π to 3π for N = 5,. n N p N [n] = n N Dr. H. Nguyen Page 27

6 4 9 Portland State University ECE 223 Discrete-Time ourier Transform Ver..4 x[n].5 Example : irst-order ilter a =.5 Example 3: irst-order ilter a =.5 ourier Transform of (.5) n u[n] X(e jω ) X(e jω ) ( o ) requency (rad/sample) 4 Portland State University ECE 223 Discrete-Time ourier Transform Ver..4 2 Dr. H. Nguyen Page 28

7 x[n] X(e jω ) Example Example : 4: irst-order irst-order ilter ilter a a =.5.5 ourier Transform of (.5) n u[n] ind tran X(e jω ) ( o ) requency (rad/sample) Portland State University ECE 223 Discrete-Time ourier Transform Ver..4 3 Portland Dr. H. Nguyen Page 29 Example 5: Workspace

8 4 3 Portland State University ECE 223 Discrete-Time ourier Transform Ver..4 4 Example 2: ourier Transform of A Pulse N = 5 Example 6: Pulse Transform for N =5 ourier Transform of p 5 [n] 8 6 P(e jω ) Portland State University ECE 223 Discrete-Time ourier Transform Ver..4 6 Dr. H. Nguyen Page 2

9 Example Example 2: ourier 7: Pulse Transform Transform of for A Pulse N = N = ourier Transform of p [n] 2 5 P(e jω ) 5 P(e jω ) Portland State University ECE 223 Discrete-Time ourier Transform Ver..4 7 Portland Dr. H. Nguyen Page 2 Example 9: Inverse of Impulse Train

10 X(e jω ) = + n= Convergence x[n]e jωn ; x[n] = X(e jω )e jωn dω Sufficient conditions of the convergence of the discrete time ourier transform of a bounded discrete-time signal: inite duration: There exits an N such that x[n] = for n > N. Absolutely summable: n= x[n] <. inite energy: n= x[n] 2 < The synthesis equation always converges There is no Gibb s phenomenon Dr. H. Nguyen Page 22

11 Properties Periodicity Linearity Time Shifting requency Shifting Conjugation a x [n] + a 2 x 2 [n] X(e j(ω+l) ) = X(e jω ) x[n n ] e jω n x[n] x [n] a X (e jω ) + a 2 X 2 (e jω ) e jωn X(e jω ) X(e j(ω ω ) ) X (e jω ) Conjugate Symmetry If x[n] is real, then X(e jω ) = X (e jω ) Differencing x[n] x[n ] ( e jω )X(e jω ) This is similar to a continuous-time derivative. Dr. H. Nguyen Page 23

12 Accumulation n m= x[n] Properties (Cont d) e jω X(ejω ) + πx(e j ) This is similar to a continuous-time integration. l= δ(ω l) Time Reversal x[ n] X(e jω ) requency Differentiation Parseval s Relation Convolution + n= n x[n] y[n] = x[n] h[n] x[n] 2 = j dx(ejω ) dω X(e jω 2 dω Y (e jω ) = X(e jω )H(e jω ) Multiplication x[n] w[n] X(e ju )W (e j(ω u) )du Dr. H. Nguyen Page 24

13 DT ourier Transform: Summary X(e jω ) is a periodic function of ω with a fundamental period of Two discrete-time complex exponentials with frequencies that differ by a multiple of are equal: e jωn = e j(ω+2lπ)n The highest perceivable discrete-time frequency is π radians per sample The lowest perceivable discrete-time frequency is radians per sample The analysis equation may or may not converge, the synthesis equation always converges The energy of the signal in the time-domain is proportional to energy of the DTT in an interval of The DTT shares many of the properties of CTT and ourier series Dr. H. Nguyen Page 25