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Phil ECE-501 Schniter January 16, 2008 ECE-501 Phil Schniter January 16, 2008 Introduction: Goal: Transmit a message from one location to another. When message is. . . continuous waveform analog comm (e.g., FM radio), sequence of numbers digital comm (e.g., mp3 le), though the sequence of numbers might represent a continuous waveform (as in the case of mp3 audio). Typical communication media: twisted pair wire (e.g., telephoneA ) coaxial cable (e.g., TVA,D , dataD ) ber optic cable (e.g., ethernetD ) EM waves (e.g., cellular phonesA,D , WiFiD , TVA,D ) water waves (e.g., underwater networkA,D ) power linesA,D compact discD hard driveD magnetic tapeA,D where A = analog and D = digital. Note that, whether the message signal is discrete-time or continuous-time, the transmitted signal is continuous-time! Analog Communication: analog message modulator transmitted signal channel received signal demessage modulator recovered Perfect recovery is impossible in the presence of noise! Digital Communication: binary message 010110 coder/ mapper symbol sequence 3,-1,1,-3 pulse shaper baseband signal baseband signal modulator passband signal channel received passband debaseband modulator received received baseband sampler received sequence equalizer de-mapper message 3.1,-0.9,1.1,-2.9 /decoder 010110 equalized sequence recovered A digital message is converted to an analog message coding and pulse-shaping, and then transmitted using analog modulation. To recover the message, the received signal is demodulated, sampled, and digitally processed. Perfect recovery is possible even in the presence of noise! 1 2 ECE-501 Phil Schniter January 16, 2008 ECE-501 Phil Schniter January 16, 2008 Preview of Comm System Components: Modulator: Translates baseband analog signal to passband : 1 1/2 1 2T 1 2T Coder/Mapper: Coder transforms sequence of message bits into an error-resiliant sequence of coded bits. Mapper transforms coded bits into discrete symbols. Ex: If the symbol alphabet is { 3, 1, 1, 3} and the symbol mapping is bits 00 01 10 11 letter a b c d . . . 1/T f fc fc f where fc is the carrier frequency. There are two principal motivations for doing this: 1. Often we want to communicate several signals simultaneously (e.g., TV, radio, voice). It s di cult or impossible to do this if they overlap in frequency! 2. Wireless EM transmission/reception is much easier at higher frequencies, since need antenna length > 10 . ( = fcc is wavelength and c=3e8 m/s speed of light.) system VHF (TV) UHF (TV) cellular WiFi transmission band 30 300 MHz 0.3 3 GHz 824 960 MHz 2.4 GHz /10 1 0.1 m 10 1 cm 3 cm 1 cm be transmitted via symbol 3 -1 , then ASCII text would 1 -3 ASCII code symbol sequence 01 10 00 01 -1 1 -3 -1 01 10 00 10 -1 1 -3 1 01 10 00 11 -1 1 -3 3 01 10 01 00 -1 1 -1 -3 . . . . . . . Notice that practical antenna length determines where di erent signal types can be transmitted. 3 4 ECE-501 Phil Schniter January 16, 2008 ECE-501 Phil Schniter January 16, 2008 Pulse Shaper: Converts symbol sequence into a continuous waveform. In linear modulation schemes, the time-n symbol s[n] scales a nT -delayed version of pulse p(t): y(t) = n Preliminaries (Ch.2): Fourier Transform (FT): De nition: W (f ) = w(t) = Properties: Linearity: F {c1 w1 (t) + c2 w2 (t)} = c1 W1 (f ) + c2 W2 (f ). Real-valued w(t) t s[n]p(t nT ) baseband signal w(t)e j2 f t dt = F {w(t)} W (f )ej2 f t df = F 1 {W (f )}. T = symbol period Ex: Say symbol sequence is [1, 3, 1, 1, 3]. Then p(t) : T 2T 3T 4T 5T 6T 3 s[n]p(t nT ) for n = 0, ..., 4 t p(t T ) : 1 t p(t 2T ) : conjugate symmetric W (f ) |W (f )| symmetric around f = 0. Bandwidth : bandpass signal: 1 1 2 t y(t) = p(t 3T ) : 3 n s[n]p(t nT ) |W (f )| 1 -power 2 0 dB BW lowpass signal: |W (f )| t p(t 4T ) : 1 3 dB f single-sided BW t 1 t T 2T 3T 4T 5T 6T 99%-power BW absolute BW double-sided BW f 5 6 ECE-501 Phil Schniter January 16, 2008 ECE-501 Phil Schniter January 16, 2008 Dirac Delta (or continuous impulse ) ( ): An in nitely tall and thin waveform with unit area: (t) limit Frequency-Domain Representation of Sinusoids: from Notice the sifting property that F 1 { (f fo )} = t (f fo )ej2 f t df = ej2 fo t . Thus, Euler s equations cos(2 fo t) = sin(2 fo t) = 1 j2 fo t e + 1 e j2 fo t 2 2 1 1 j2 fo t e 2j e j2 fo t 2j that s often used to kick a system and see how it responds. Key properties: 1. Sifting: and the Fourier transform pair ej2 fo t (f fo ) imply that w(t) (t q)dt = w(q). F {cos(2 fo t)} = Often we draw this as 2. Time-domain impulse (t) has a at spectrum: F { (t)} = (t)e j2 f t dt = 1 (for all f ). F {sin(2 fo t)} = 1 (f fo ) + 1 (f + fo ) 2 2 1 1 (f fo ) 2j (f + fo ). 2j |F {cos(2 fo t)}| = |F {sin(2 fo t)}| 1 2 3. Freq-domain impulse (f ) corresponds to a DC waveform: F 1 { (f )} = (f )ej2 f t df = 1 (for all t). fo fo f 7 8 ECE-501 Phil Schniter January 16, 2008 ECE-501 Phil Schniter January 16, 2008 Frequency Domain via MATLAB: Fourier transform requires evaluation of an integral. What do we do if we can t de ne/solve the integral? 1 1. Generate (rate- Ts ) sampled signal in MATLAB. Linear Time-Invariant (LTI) Systems: An LTI system can be described by either its impulse response h(t) or its frequency response H(f ) = F {h(t)}. in time : domain in frequency : domain impulse (t) LTI system LTI system h(t) impulse response frequency response 2. Plot magnitude of Discrete Fourier Transform (DFT) using plottf.m (from course webpage). Square-wave example: 1 0.5 amplitude 0 at 1 spectrum H(f ) Input/output relationships: Time-domain: Convolution with impulse response h(t) 0 0.2 0.4 0.6 0.8 1 time 1.2 1.4 1.6 1.8 2 magnitude f = 10; t_max = 2; Ts = 1/1000; t = 0:Ts:t_max; x = sign(cos(2*pi*f*t)); plottf(x,Ts); !0.5 !1 x(t) h(t) y(t) y(t) = h(t) x(t) = 1.4 1.2 1 0.8 0.6 0.4 0.2 0 !500 !400 !300 !200 !100 0 frequency 100 200 300 400 500 h(t )x( )d Freq-domain: Multiplication with freq response H(f ) X(f ) H(f ) Y (f ) Y (f ) = H(f )X(f ) Noise-wave example: 3 2 amplitude 1 0 !1 !2 !3 0 0.1 0.2 0.3 0.4 0.5 time 0.6 0.7 0.8 0.9 1 Linear Filtering: Freq-domain illustration of LPF, BPF, and HPF: X(f ) f BPF !400 !300 !200 !100 0 frequency 100 200 300 400 500 t_max = 1; Ts = 1/1000; x = randn(1,t_max/Ts); plottf(x,Ts); magnitude 0.08 H(f ) LPF = f f Y (f ) f f f 0.06 0.04 0.02 0 !500 f HPF Notice that plottf.m only plots frequencies f 9 1 1 [ 2Ts , 2Ts ). f f 10 ECE-501 Phil Schniter January 16, 2008 ECE-501 Phil Schniter 1 -sampled Ts January 16, 2008 Lowpass Filters: Ideal non-causal LPF (using sinc(x) := 1 |f | B H(f ) = 0 |f | > B 1 F In MATLAB, generate sin( x) ): x LPF impulse response via h = firls(Lf, [0,fp,fs,1], [G,G,0,0])/Ts; where. . . G 0 f 1/(2Ts ) h(t) = 2B sinc(2Bt) 2B Lf+1 = impulse response length {G, G}, {0, 0} = corresp. magnitude pairs {0, fp}, {fs, 1} = normalized freq pairs H(f ) h(t) B 0 B f 1 B 1 2B 0 1 2B 1 B t 0 fp fs 1 Ideal LPF with group-delay to : e j2 f t0 H(f ) = 0 1 The commands firpm and fir2 have the same interface, but yield slightly di erent results (often worse for our apps). In MATLAB, perform ltering on t_max = 3; Ts = 1/1000; x = randn(1,t_max/Ts); h = firls(100,[0,0.2,0.4,1],[1,1,0,0])/Ts; y = Ts*filter(h,1,x); subplot(3,1,1); plottf(x,Ts, f ); ylabel( |X(f)| ) subplot(3,1,2); plottf(h,Ts, f ); ylabel( |H(f)| ) subplot(3,1,3); plottf(y,Ts, f ); ylabel( |Y(f)| ) |f | > B |f | B 2B h(t) = 2B sinc 2B(t t0 ) h(t) 1 2B F 1 -sampled Ts signal x via |H(f )| y = Ts*filter(h,1,x); or y = Ts*conv(h,x); 0.2 0.15 |X(f)| 0.1 0.05 0 !500 1.5 1 0.5 0 !500 0.2 0.15 |Y(f)| 0.1 0.05 0 !500 !400 !300 !200 !100 0 frequency 100 200 300 400 500 x 10 !3 B 0 B f t0 t !400 !300 !200 !100 0 frequency 100 200 300 400 500 A causal linear-phase LPF with group-delay to : 1 |H(f)| !400 !300 !200 !100 0 frequency 100 200 300 400 500 |H(f )| 2B h(t) 1 2B B 0 B f t0 2t0 t symmetry around center yields linear phase but MATLAB can give better causal linear-phase LPFs. . . 11 Important: The routines firls,firpm,fir2 generate causal linear-phase lters with group delay = Lf samples. Thus, the 2 ltered output y will be delayed by Lf samples relative to x. 2 12

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