Department of Electrical Engineering University of Arkansas ELEG 5173L Digital Signal Processing Ch. 3 Discrete-Time Fourier Transform Dr. Jingxian Wu [email protected] 2 OUTLINE • The Discrete-Time Fourier Transform (DTFT) • Properties • DTFT of Sampled Signals • Upsampling and downsampling 3 DTFT • Discrete-time Fourier Transform (DTFT) X () x(n)e jn n – (radians): digital frequency • Review: Z-transform: X (z) x(n)zn n0 j X () X (z) | j – Replace z with e . ze • Review: Fourier transform: X () x(t)e jt – (rads/sec): analog frequency 4 DTFT • Relationship between DTFT and Fourier Transform – Sample a continuous time signal x a ( t ) with a sampling period T xs (t) xa (t) (t nT ) xa (nT ) (t nT ) n n – The Fourier Transform of ys (t) jt jnT X s () xs (t)e dt xa (nT)e n – Define: T • : digital frequency (unit: radians) • : analog frequency (unit: radians/sec) – Let x(n) xa (nT) X () X s T 5 DTFT • Relationship between DTFT and Fourier Transform (Cont’d) – The DTFT can be considered as the scaled version of the Fourier transform of the sampled continuous-time signal jt jnT X s () xs (t)e dt xa (nT)e n x(n) x (nT) T a jn X () X s x(n)e T n 6 DTFT • Discrete Frequency – Unit: radians (the unit of continuous frequency is radians/sec) – X ( ) is a periodic function with period 2 j2 n jn j2n jn X ( 2 ) x(n)e x(n)e e x(n)e X () n n n – We only need to consider for • For Fourier transform, we need to consider 1 – f T 2 2T 1 – f T 2 2T 7 DTFT • Example: find the DTFT of the following signal – 1. x(n) (n) n – 2. x(n) u(n), 1 8 DTFT • Example – Find the DTFT of x(n) u(n) u(n N) 9 DTFT • Example – Find the DTFT of the following signal x(n) |n|, 1 • Existence of DTFT – The DTFT of x(n) exists if x(n) satisfies | x(n) | n 10 DTFT • Inverse DTFT 1 jn x(n) X ()e d 2 2 • Example – Find the inverse DTFT of ( 0 ) 11 DTFT • Example – Find the DTFT of cos(0n ) 12 OUTLINE • The Discrete-Time Fourier Transform (DTFT) • Properties • DTFT of Sampled Signals • Upsampling and downsampling 13 PROPERTIES • Periodicity X ( 2) X() • Linearity x (n) X () – If x1(n) X1() 2 2 – Then x1(n) x2(n) X1() X2 () 14 PROPERTIES • Time shifting – If x(n) X () – Then jn0 x(n n0 ) X ()e • Frequency shifting – If x(n) X () – Then j0n e x(n) X ( 0 ) 15 PROPERTIES • Differentiation in Frequency – If x(n) X () – Then dX () nx(n) j d d nx(n) z X (z) – Review dz • Example n – Find the DTFT of n u(n) | |1 16 PROPERTIES • Convolution – If y(n) h(n) x(n) – Then Y() H()X () • Example – Find the frequency response of the system y(n) x(n n0 ) 17 PROPERTIES • Example n 1 – A LTI system with impulse response h(n) u(n) n 2 if the input is 1 x(n) u(n) 3 Find the output 18 PROPERTIES • Example – The frequency response of an ideal low pass filter is 1, 0 c H() 0, c find the impulse response. (1) Assume c 0 . 5 . If the sampling rate of the input signal is 1 KHz, what is the cutoff frequency for the analog signal? (2) If the sampling rate of the input signal is 2 KHz, what is the cutoff frequency of the analog signal? 19 PROPERTIES • Modulation – If y(n) x1(n)x2 (n) – Then 1 2 Y() X1( p)X 2 ( p)dp 2 0 20 OUTLINE • The Discrete-Time Fourier Transform (DTFT) • Properties • Application: DTFT of Sampled Signals • Upsampling and downsampling 21 APPLICATION: SAMPLING THEOREM • Sampling theorem: time domain – Sampling: convert the continuous-time signal to discrete-time signal. xa (t) p(t) (t nT ) n sampling period xs (t) xa (t)p(t) 22 APPLICATION: SAMPLING THEOREM • Sampling theorem: frequency domain – Fourier transform of the impulse train • impulse train is periodic Fourier series 1 p(t) (t nT ) 1 e jnst 2 s n T n T • Find Fourier transform on both sides 2 P() ( ns ) T n • Time domain multiplication Frequency domain convolution 1 x(t) p(t) X () P() 2 a 1 2 xs (t) x(t) p(t) X s () X a n T n T 23 APPLICATION: SAMPLING THEOREM • Sampling theorem: frequency domain – Sampling in time domain Repetition in frequency domain 24 APPLICATION: SAMPLING THEOREM • Sampling theorem – If the sampling rate is twice of the bandwidth, then the original signal can be perfectly reconstructed from the samples. s 2B s 2B s 2B s 2B aliasing Department of Engineering Science Sonoma State University 25 APPLICATION: SAMPLING THEOREM • Analog signal x a ( t ) v.s. sampled signal x s ( t ) v.s. discrete-time x(n) signal x(n) xa (nT ) Discrete-time signal Samples of continuous-time signal x (t) (t nT ) x (nT ) (t nT ) x(n) (t nT ) xs ( t) a a n n n xs (t) x(n) (t nT ) n 26 APPLICATION: SAMPLING THEOREM • Fourier transform of sampled signal jt jnT X s () xs (t)e dt x(n)e n • DTFT of discrete-time signal X () x(n)e jn n • Relationship X () X s T DTFT of discrete-time signal Fourier transform of continuous-time signal 27 APPLICATION: SAMPLING THEOREM • Relationship among X (), X s (), X a () – X () X s T 1 2 X s () X a n T n T 1 2 X () X a n T n T T T 28 APPLICATION: SAMPLING THEOREM • Example – Consider the analog signal x a ( t ) with Fourier transform shown in the figure. The signal has a one-sided bandwidth 5 Hz. The signal is sampled with a sampling period T . 25ms X () Draw X s ( ) and 1 29 APPLICATION: SAMPLING THEOREM • Reconstruction of sampled signal – If there is no aliasing during sampling, the analog signal can be reconstructed by passing the sampled signal through an ideal low pass filter with cutoff frequency B T, | | s H() 2 0, otherwise t h(t) sin c T xs (t) x(n) (t nT ) n t nT xˆa (t) xs (t) h(t) x(n)sinc n T 30 OUTLINE • The Discrete-Time Fourier Transform (DTFT) • Properties • Application: DTFT of Sampled Signals • Upsampling and downsampling 31 UPSAMPLING AND DOWNSAMPLING • Sampling rate conversion – In many applications, we may have to change the sampling rate of a signal as the signal undergoes successive stage of processing. • E.g. Rate h1(n) h2 (n) conversion h3(n) T Sampling period T1 T1 2 – If h ( n ) is a low pass filter with the cut-off frequency being 2 0 u then what is the largest value of T 2 ? 32 UPSAMPLING AND DOWNSAMPLING • Sampling rate conversion – Sampling rate conversion can be directly performed in the digital domain – Downsampling (decimation) • Reduce the sampling frequency (less samples per unit time) – Upsampling (interpolation) • Increase the sampling frequency (more samples per unit time) 33 UPSAMPLING AND DOWNSAMPLING • Downsampling (decimation) – Assume a digital signal,x ( n ) , is obtained by sampling an analog signal x a ( t ) at a sampling period T (or a sampling frequency 1/T). x(n) xa (nT ) – The signal can also be sampled with a sampling period MT (or a lower sampling frequency 1/(MT) ) xd (n) xa (nMT) – Relationship between x ( n ) and xd (n) xd (n) x(nM ) – x d ( n ) is the downsampled (or decimated) version of x(n) • can be obtained by picking one sample out of every M consecutive samples of x(n) decimation. x(n) [x(0), x(1), x(2), x(3), x(4), x(5), x(6), x(7), x(8),] x (n) [x(0), x(3), x(6), ,] M 3 d 34 UPSAMPLING AND DOWNSAMPLING • Downsampling (decimation): frequency domain – Assume all the sampling rates are higher than the Nyquist sampling rate – The DTFT of x(n) with a sampling period T 1 X () X a T T – The DTFT of x d ( n ) with a sampling period MT 1 X d () X a MT MT 1 X d () X M M 35 UPSAMPLING AND DOWNSAMPLING • Example – Consider the frequency response of an ideal low pass filter 1, H() 2 2 0, , 2 2 • Determine h(n) • If we downsample h(n) with a factor of M = 2. Find the downsampled response in both the time and frequency domains. 36 UPSAMPLING AND DOWNSAMPLING • Upsampling (interpolation) – Assume a digital signal,x ( n ) , is obtained by sampling an analog signal x a ( t ) at a sampling period T (or a sampling frequency 1/T). x(n) xa (nT ) – Upsampling by a factor of L • Inserting L-1 zeros between the original samples x(n) [x(0), x(1), x(2), x(3),] L 3 xu (n) [x(0),0,0, x(1),0,0, x(2),0,0, x(3),] T – The effective sampling period of the upsampled signal (sampling L rate: L ) T xu (n) xu (kL) x(k) if n / L k is an integer xu (n) 0, if n / L integer 37 UPSAMPLING AND DOWNSAMPLING • Upsampling (interpolation): frequency domain – The DTFT of x u ( n ) with a sampling period T/L jn jkL X u () xu (n)e x(k)e X (L) n k • The DTFT of X ( ) is compressed in the frequency domain 38 UPSAMPLING AND DOWNSAMPLING • Example – A discrete pulse is given by x(n) = u(n)-u(n-4).