6.003: Signals and Systems Lecture 18 April 13, 2010

Transcription

1 6.003: Signals and Systems Lecture 8 April 3, : Signals and Systems Discrete-Time Frequency Representations Signals and/or Systems Two perspectives: feedback and control (focus on systems) X + controller plant sensor Y Is the system stable? signal processing (focus on signals) target image April 3, 200 Learn about target (signal) from the image (signal). Fourier methods are especially useful in signal processing. Historical Perspective Broad range of CT signal-processing problems: audio radio (noise/static reduction, automatic gain control, etc.) telephone (equalizers, echo-suppression, etc.) hi-fi (bass, treble, loudness, etc.) television (brightness, tint, etc.) radar and sonar (sensitivity, noise suppression, object detection)... Increasing important applications of DT signal processing: MP3 JPEG MPEG MRI... Signal Processing: Acoustical Mechano-acoustic components to optimize frequency response of loudspeakers: e.g., bass-reflex system. driver reflex port Signal Processing: Acoustico-Mechanical Passive radiator for improved low-frequency preformance. Signal Processing: Electronic The development of low-cost electronics enhanced our ability to alter the natural frequency responses of systems. 0 driver passive radiator Magnitude (db) Frequency (Hz) Eight drivers faced the wall; one pointed faced the listener. Electronic equalizer compensates for limited frequency response.

2 6.003: Signals and Systems Lecture 8 April 3, 200 Signal Processing Signal Processing Modern audio systems process sounds digitally. Modern audio systems process sounds digitally. AVSS(REF) V REFM V REFP V RFILT AVDD AVSS DVDD DVSS AIRP AIRM RIA RIB Voltage Reference Analog Supplies Digital Supplies x(t) x[n] y[n] A/D DT filter D/A y(t) Texas Instruments TAS3004 AILP AILM LIA LIB AIRP AIRM 24-Bit Stereo ADC AILP Analog Register Output Format Logic SDOUT0 2 channels 24 bit ADC, 24 bit DAC 48 khz sampling rate 00 MIPS $7.70 ($4.8 in bulk) ALLPASS IPA GPI5 GPI4 GPI3 GPI2 GPI GPI0 CS SDA SCL ler I 2 C L+R AILM 24-Bit Stereo DAC 32-Bit Audio Signal Processor VCOM AOUTL AOUTR L+R SDOUT2 SDOUT Courtesy of Texas Instruments. Used with permission. PWR_D RESET TEST L R SDI SDI2 SDATA LRCLK/O SCLK/O IFM/S 32-Bit Audio Signal Processor OSC/CLK Select CLKSEL XTALI/ MCLK XTALO MCLKO PLL CAP_PLL Figure -. TAS3004 Block Diagram and Frequency Response Today: frequency representations for DT signals and systems. Complex Geometric Sequences Complex geometric sequences are eigenfunctions of DT LTI systems. n Find response of DT LTI system (h[n]) to input x[n] =z. y[n] =(h x)[n] = h[k]z n k = z n h[k]z k = H(z) z n. k= k= Complex geometrics (DT): analogous to complex exponentials (CT) z n h[n] n H(z) z e st h(t) H(s) e st Rational System Functions A system described by a linear difference equation with constant coefficients system function that is a ratio of polynomials in z. DT Vector Diagrams Factor the numerator and denominator of the system function to make poles and zeros explicit. (z 0 q 0 )(z 0 q )(z 0 q 2 ) Example: H(z 0 )=K (z0 p 0 )(z 0 p )(z 0 p 2 ) y[n 2]+3y[n ]+4y[n] =2x[n 2]+7x[n ]+8x[n] 2z 2 +7z z +8z 2 (z) H(z) = z 2 +3z +4 = +3z +4z 2 D(z) z 0 q 0 z 0 z 0 z-plane q q 0 0 Each factor in the numerator/denominator corresponds to a vector from a zero/pole (here q 0 )to z 0, the point of interest in the z-plane. Vector diagrams for DT are similar to those for CT. 2

3 6.003: Signals and Systems Lecture 8 April 3, 200 DT Vector Diagrams Value of H(z) at z = z 0 can be determined by combining the contributions of the vectors associated with each of the poles and zeros. (z 0 q 0 )(z 0 q )(z 0 q 2 ) H(z 0 )= K (z0 p 0 )(z 0 p )(z 0 p 2 ) DT Frequency Response Response to eternal sinusoids. Let x[n] = cos Ω 0 n (for all time): ( ) ( ) x[n] = e jω 0n + e jω 0n = 2 2 z 0 n n + z The magnitude is determined by the product of the magnitudes. (z 0 q 0 ) (z 0 q ) (z 0 q 2 ) H(z 0 ) = K (z0 p 0 ) (z 0 p ) (z 0 p 2 ) The angle is determined by the sum of the angles. H(z 0 )= K + (z 0 q 0 )+ (z 0 q )+ (z 0 p 0 ) (z 0 p ) where z 0 = e jω 0 and z = e jω 0. The response to a sum is the sum of the responses: ( ) n n y[n] = H(z 0 ) z H(z ) z ( ) = H(e jω 0 ) e jω 0n + H(e jω 0 ) e jω 0n 2 Conjugate Symmetry For physical systems, the complex conjugate of H(e jω ) is H(e jω ). The system function is the Z transform of the unit-sample response: H(z) = h[n]z n n= where h[n] is a real-valued function of n for physical systems. H(e jω )= h[n]e jωn n= ( ) H(e jω )= h[n]e jωn H(e jω ) n= DT Frequency Response Response to eternal sinusoids. Let x[n] = cos Ω 0 n (for all time), which can be written as x[n] = ( ) e jω 0n + e jω 0n. 2 Then ( ) y[n] = H(e jω 0 )e jω 0n + H(e jω 0 )e jω 0n 2 { } =Re H(e jω 0 )e jω 0n { } =Re H(e jω 0 ) e j H(e jω 0 ) e jω 0n { } = H(e jω 0 ) Re e jω 0n+j H(e jω 0 ) ( ) y[n] = H(e jω 0 ) cos Ω 0 n + H(e jω 0 ) Frequency Response The magnitude and phase of the response of a system to an eternal cosine signal is the magnitude and phase of the system function evaluated on the unit circle. cos(ωn) H(z) ( ) H(e jω ) cos Ωn + H(e jω ) Comparision of CT and DT Frequency Responses CT frequency response: H(s) on the imaginary axis, i.e., s = jω. DT frequency response: H(z) on the unit circle, i.e., z = e jω. ω s-plane z-plane H(e jω )= H(z) z=e jω σ H(jω) jω ) H(e

4 6.003: Signals and Systems Lecture 8 April 3, 200 Periodicity of DT Frequency Responses Check Yourself DT frequency responses are periodic functions of Ω, with period 2. If Ω 2 =Ω +2k where k is an integer then H(e jω 2 ) =H(e j(ω +2k) )=H(e jω e j2k )=H(e jω ) What kind of filtering corresponds to the following? z-plane The periodicity of H(e jω ) results because H(e jω ) is a function of e jω, which is itself periodic in Ω. Thus DT complex exponentials have many aliases. e jω 2 = e j(ω +2k) = e jω e j2k = e jω Because of this aliasing, there is a highest DT frequency: Ω=.. high pass 2. low pass 3. band pass 4. band stop (notch) 5. none of above DT Fourier series represent DT signals in terms of the amplitudes and phases of harmonic components. x[n] = a k e jkω 0n The period of all harmonic components is the same. There are distinct complex exponentials with period. If e jωn is periodic in then e jωn = e jω(n+) = e jωn e jω and e jω must be, and Ω must be one of the th roots of. Example: =8 z-plane There are distinct complex exponentials with period. These can be combined via Fourier series to produce periodic time signals with independent samples. Example: periodic in =3 n 3 samples repeated in time 3 complex exponentials Example: periodic in =4 n 4 samples repeated in time 4 complex exponentials DT Fourier series represent DT signals in terms of the amplitudes and phases of harmonic components. x[n] =x[n + ] = a k e jkω 0n ; Ω 0 = 2 equations (one for each point in time n) in unknowns (a k ). Example: =4 x[0] e j e j 2 0 e j e j a 0 j 2 0 j 2 j 2 2 j 2 3 x[] e e e e a = x[2] j j 2 2 j j e e e e a 2 x[3] e j e j 2 3 e j e j a 3 4

5 6.003: Signals and Systems Lecture 8 April 3, 200 DT Fourier series represent DT signals in terms of the amplitudes and phases of harmonic components. Solving these equations is simple because these complex exponentials are orthogonal to each other. x[n] = x[n + ] = a k e jkω 0n ; Ω 0 = 2 jω 0 kn jω 0 ln jω = 0 (k l)n e e e n=0 n=0 { ; k = l equations (one for each point in time n) in n unknowns (a k ). = e jω 0 (k l) =0 ; k = l Example: =4 x[0] a0 x[] j j a = x[2] a 2 x[3] j j a 3 e jω 0 (k l) = δ[k l] We can use the orthogonality property of these complex exponentials to sift out the Fourier series coefficients, one at a time. Assume x[n] = a k e jkω 0n Multiply both sides by the complex conjugate of the l th harmonic, and sum over time. x[n]e jlω 0n = a k e jkω 0n e jlω 0n = a k e jkω 0n e jlω 0n n=0 n=0 n=0 = a k δ[k l] = a l a k = x[n]e jkω 0n n=0 Since both x[n] and a k are periodic in, the sums can be taken over any successive indices. otation. If f[n] is periodic in, then + f[n] = f[n] = f[n] = = f[n] n=0 n= n=2 n=<> a k = a k+ = x[n]e jω 0n n=<> x[n]= x[n + ] = a k e jkω 0n k=<> ; Ω 0 = 2 ( analysis equation) ( synthesis equation) DT Fourier series have simple matrix interpretations. x[n] = x[n +4] = jk n a k e jkω 2 0n = a k e 4 = a k j kn k=<4> k=<4> k=<4> x[0] a 0 x[] j j a = x[2] a 2 x[3] j j a 3 a k = a k+4 = x[n]e jkω 0n = e jk2 n = x[n]j kn n=<4> n=<4> n=<4> a 0 x[0] j j x[] a = a 2 x[2] a 3 j j x[3] Discrete-Time Frequency Representations Similarities and differences between CT and DT. DT frequency response vector diagrams (similar to CT) frequency response on unit circle in z-plane (jω axis in CT) DT Fourier series represent signal as sum of harmonics (similar to CT) finite number of periodic harmonics (unlike CT) finite sum (unlike CT) These matrices are inverses of each other. 5

6 MIT OpenCourseWare Signals and Systems Spring 200 For information about citing these materials or our Terms of Use, visit: