II.B Baseband Transmission (Reception & Applications)
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1 II.B Baseband Transmission (Reception & Applications) Digital Baseband Reception Matched filter Definition Implementation (correlator) Application to digital baseband signal M-ary Baseband Reception Brief Review of Probability and Noise Concepts Basic pdf definitions White noise Narrowband noise Bit Error Rate Error probability Threshold definition Application to the binary matched filter detector Examples M-ary baseband Performance Application to the CD Format Speech range D/A converter issues Oversampling Noise shaping & Dither effects 10/3/03 EC500MPF/FallFY04 4
2 10) Digital Baseband Reception s(t) V V T b t 1 n T b compare to zero Goal: to recover s(t) from potentially noisy received signal use a filter to decrease the effect of noise Generic filter does not take advantage of known signal shape transmitted better result obtained when using that information matched filter 10/3/03 EC500MPF/FallFY04 43
3 Matched Filter s(t) + n(t) H (f) T s 0 (t) + n 0 (t) Definition: a matched filter is a linear filter which minimizes the output signal to noise ratio (SNR) at time T, where is defined as: s n 0 0 T t j ft S f H f e df n S f H f df Goal: find H(f) which minimizes Proof: Use Schwartz s inequality which states: A x B x dx A x dx B x dx equality holds only when A(n) = KB * (x) K real constant 10/3/03 EC500MPF/FallFY04 44
4 j ft S f H f e df n S f H f df A B f f j ft S j ft S f H f e df H f Sn f e df S n S S S S f f n n f f j ft df H f Sn f e df j ft n df H f S f e df n S f df S f n H f S f df S f H f df n n S f df S max obtained when f S S f f * j ft n n 10/3/03 EC500MPF/FallFY04 45 n * S f H f K e S f KH f S f e j ft
5 When n(t) is white noise H f KS f e * j ft ht KsT t S f n n Example: 1 s(t) h(t) T t t 10/3/03 EC500MPF/FallFY04 46
6 Matched Filter Implementation (correlator) s(t) + n(t) H (f) y(t) T yt st nt ht s nht d h KsT s ns d T 10/3/03 EC500MPF/FallFY04 47
7 Matched Filter Detector for Digital Baseband s(t)... T b s(t) T b 0. dt y(t) 10/3/03 EC500MPF/FallFY04 48
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10 M-ary Baseband Reception We can transmit more than two symbols Example: 4-ary baseband communications How to apply binary result to M-ary case? 10/3/03 EC500MPF/FallFY04 51
11 11) Brief Review of Probability and Noise Concepts Probability distribution function F (x) = Probability density function f (x) Expected value E (x) = Variance Basic pdfs x (1) Uniform pdf f (x) x f (x) = 10/3/03 EC500MPF/FallFY04 5
12 () Gaussian pdf f (x) x f (x) = 10/3/03 EC500MPF/FallFY04 53
13 White Noise x(n) is random with a constant PSD G x (f) N 0 / f Quadratic noise components Narrowband Noise most communication systems contain bandpass filters white noise gets transformed into BP noise when noise band is small compared to center frequency f c BP noise called narrowband noise expressed as: cos c sin c j fct Re r t e wt x t ft y t ft complex function 10/3/03 EC500MPF/FallFY04 54
14 w(n) BPF v(n) white noise N 0, w H(f) f f H f L f L f H P out EV n V 10/3/03 EC500MPF/FallFY04 55
15 1) Bit Error Rate Evaluate errors made during transmission of hits Send Receive 0 0 P M P FA 1 1 Errors occur when: receive 1 when 0 is sent receive 0 when 1 is sent How to decide if you receive a 1 or 0 when transmission is noisy?? detection theory 10/3/03 EC500MPF/FallFY04 56
16 r(t)=s(t)+kw(t) N(0, w =1) 10/3/03 EC500MPF/FallFY04 57
17 Assume you have additive white noise distortion at the receiver r(t)=s i (t)+w(t), s i (t)=s 0, s 1 f (w) = f (r s 0 ) = 0 r f (r s 1 ) = For which range of r do you decide you sent a 1 (s 1 ) 0 (s 0 ) How to pick 0? 10/3/03 EC500MPF/FallFY04 58
18 Goal: To quantify likelihood of making an error in assigning bit values at the receiver. Assume transmission is noisy r n s n w n received signal signal sent s0 s1 transmission noise (random Gaussian) N 0, w Statistics of r(n) r(n) deterministic? random? mean of r(n)? 10/3/03 EC500MPF/FallFY04 59
19 Notation H 1 : receive a 1 (s 1 ) H 0 : receive a 0 (s 0 ) Correct decisions: P H s P H 1 1 s 0 0 Incorrect decisions: P H s P H 1 0 s 0 1 Overall probability of error: Pe P e when there is equal probability of sending s 0 & s 1 10/3/03 EC500MPF/FallFY04 60
20 1 Pe PH ( s) PH ( s) f( r s) dr Assume s 0 =-V & s 1 =+V Definitions: Q, erf, & erfc functions 1 u / 1 ( ) erfc x Qx e du x x u erfc( x) 1 e du 1erf ( x) 0 10/3/03 EC500MPF/FallFY04 61
21 BER for single sample detector 10/3/03 EC500MPF/FallFY04 6
22 How to select the threshold 0? Maximum likelihood detector approach Minimize the overall probability of error P e P PH ( s) Ps ( ) PH ( s) Ps ( ) e s s P w In P1 s1 s0 (More details in EC4570 ) r 0 10/3/03 EC500MPF/FallFY04 63
23 Application to Binary Matched Filter Detector r(t)=s i (t)+w(t) T b 0 dt y 1 r(t) s 1 (t) y T b compare to threshold T b 0 dt y 0 s 0 (t) Is y random or deterministic? Need statistics on y = y 1 y 0 pdf of y:? 10/3/03 EC500MPF/FallFY04 64
24 T b y r t s t dt T si t w t s1 t dt si t s1 t or s0 t b T b Tb s t s t dt w t s t dt i y y y t y t 10/3/03 EC500MPF/FallFY04 65
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26 10/3/03 EC500MPF/FallFY04 67
27 Bit error rate for matched filter detector T b 0 s () t s () t dt 0 1 E Tb Tb 1 E s0() t dt s1 () t dt /3/03 EC500MPF/FallFY04 68
28 How to compute the threshold 0? Recall for simple detector, the threshold was selected as the mid point between the two means for basic problem. How can we apply the result here? E[y s 0 ]= E[y s 1 ]= 10/3/03 EC500MPF/FallFY04 69
29 Example s 0 (t) s 1 (t) +1 1 T t +1 T t T = s Design a matched filter detector for the two signals. Find P e when the additive white noise has a power P = 10 3 w/hz. 10/3/03 EC500MPF/FallFY04 70
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31 10/3/03 EC500MPF/FallFY04 7
32 Example s 0 (t) +1 1 s 1 (t) sin(t) t 1 1 t Design a matched filter detector for the two signals. Find P e when the additive white noise has a power P e = 0.1 w/hz. 10/3/03 EC500MPF/FallFY04 73
33 10/3/03 EC500MPF/FallFY04 74
34 M-ary Baseband Performance Assume we send M = 4 different levels (B i = 0, A, A, 3A, i=0, 3) T b r(t) dt? 0 Assume additive Gaussian noise r(t)=s(t)+n(t) 10/3/03 EC500MPF/FallFY04 75
35 P(error in receiving B )= 10/3/03 EC500MPF/FallFY04 76
36 AT s AT s P(error in receiving B ) Q erfc N 4N 0 0 P(error in receiving B 1 )= P(error in receiving B 0 )= P(error in receiving B 3 )= Assume each error may occur as likely as the others P e = 10/3/03 EC500MPF/FallFY04 77
37 Assume we send M different levels, compute the overall probability of error becomes: 10/3/03 EC500MPF/FallFY04 78
38 13) Application: Compact Disk Dynamic range for audio signals Ref [3] 10/3/03 EC500MPF/FallFY04 79
39 Signal reproduction in CD Player x(n) Digital input 16 bits 44.1 KHz Digital oversampling filter X4 y(n) 8 bits KHz Noise shaper H(f) 14 bits 14 bits D/A LPF x a (n) 0 KHz 0 KHz f x(n) KHz 88. KHz f 10/3/03 EC500MPF/FallFY04 80
40 Why use oversampling? First, assume we don t use oversampling. x(r) 14 bits D/A z(t) LPF H r (j) y(t) 7 analog output Input/output relationship for a unipolar D/A (3 bits) converter x 1 (t) z(t) (p. 331) t t Ideal D/A Converter Practical D/A Converter x(n) x 1 (t) H z (f) z(t) h z (t) 1 T 10/3/03 EC500MPF/FallFY04 81 t
41 D/A Converter Output Expression 1 x t x k t kt k z t x t h t 1 Z j X j H z 1 z H z (j) T jt Hz e dt 0 1 j e jt 1 jt e sin T X 1 (j) s s 3 s s / T Z(j) s s 3 s s s 3 s 10/3/03 EC500MPF/FallFY04 8
42 Need for LPF filter? to smooth out output steps to undo distortion added by D/A converter Z(j) s s H r (j) H r (j) 10/3/03 EC500MPF/FallFY04 83
43 x(n) 16 bits 44.1 KHz Digital oversampling filter X4 y(n) Noise shaper H s (f) 14 bits D/A LPF a(n) KHz Oversampling in the Time Domain 010 input to upsampler by n output to upsampler by n output of FIR filter n x(n) 4 y (n) FIR filter y(n) 10/3/03 EC500MPF/FallFY04 84
44 X a (f) f (KHz) X(f) without oversampling f (KHz) Y 1 (f) with oversampling f (KHz) Y(f) Y 1 (f) after FIR filter f (KHz) A(f) after analog LPF f (KHz) Advantages of Oversampling 10/3/03 EC500MPF/FallFY04 85
45 Example: Assume 1) you have an analog signal which spans [0 0KHz], ) the D/A converter has a sampling frequency f s =176.4KHz. Determine the characteristics (order and cutoff frequency) for the anti-imaging Butterworth type filter which satisfy the following specifications: 1) Image frequencies must be attenuated by at least 40dB ) Signal components may be altered by a maximum of 0.5dB H( f) 1 f / 1 f c n 1/ 10/3/03 EC500MPF/FallFY04 86
46 10/3/03 EC500MPF/FallFY04 87
47 Noise Shaping Filter Goal: to decrease noise in the audio band noise shaping characteristic noise level noise level without shaping 0 88 f KHz 10/3/03 EC500MPF/FallFY04 88
48 Dither Figure.11 - Coding of Dithered Signal Figure.1 - Fourier Transform of Dithered Signal 10/3/03 EC500MPF/FallFY04 89
49 Dither & noise shaping effects Poh1man Fig. 6-7 An example of noise shaping showing a 1 khz sinewave with -90 db amplitude; measurements are made with a 16 khz lowpass filter. A. Original 0 bit recording. B. Truncated 16 bit signal. C. Dithered 16 bit signal. D. Noise shaping preserves information in lower 4 bits. Ref [4,5] 10/3/03 EC500MPF/FallFY04 90
50 CD Section References: [1] S. Mitra, Digital Signal Processing, McGraw-Hill, [] K. Pohlmann, A. Red, Compact Disk Handbook,, 199. [3] H. Nakajima and H. Ogawa, Compact Disk Technology, IOS Press, 199. [4] B. Evans, Real Time Digital Signal Processing Lab, Fall 003 (lecture 10 notes). [5] K. Pohlmann, Principles of Digital Audio, McGraw- Hill, /3/03 EC500MPF/FallFY04 91
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