6.02 Fall 2010 Lecture #16. Reminders FD sharing Fourier Series and Pictures Mod and Demod New issue, sine mod -> cosine demod Channel Delay

Transcription

1 6.02 Fall 2010 Lecture #16 Reminders FD sharing Fourier Series and Pictures Mod and Demod New issue, sine mod -> cosine demod Channel Delay

2 x 1 [n] Frequency Domain Sharing Big Picture Mod DeMod y 1 [n] x 2 [n]... Mod + X Channel Y DeMod y 2 [n]... x P [n] Mod DeMod y P [n] Questions What is Modulation (Mod in figure)? What do typical X s look like? What should the Demodulator be? What is the relation between x i [n] and y i [n] What happens if there is channel delay (NEW)

3 Modulation by Cosine Multiplication x 1 [n]... cos 1 n x[n] = P P i=1 x i[n] cos i n x P [n] cos P n

4 What does a modulated X look like? Can Fourier Series Help us understand Modulation?

5 Periodic Assumption (with period N) One Period Periodic with Period N (N even) means Only certain frequency complex exponentials x[n] can be represented with a Fourier series

6 Discrete-Time Fourier Series OR

7 Key DTFS Modulation Identity

8 Key DTFS Modulation Identity ~ / M Q.lJ.. ",.::h9/l P:re: 't '"e. i\ C'7 Et 1< Q [ "!<;J. '1.. < /(In x[n] COS Om [n] = L X[k]e J " k.s k~-k 1... n.~.,,,,, s : 1 f. ('l ').?"-i _. ej.. /. '...'.' ';'n4n+ -«:» 'tnn, J Y\Jt c,...r..-"-) p;' L-11 k - Z11. J! -::,-:~~ N tj +e.. ~ ilk =: N" Q(V1 - NM Ffe~telE 1 K 1 K 2 L X[k]ejcnk+n.n)n + 2 L X[klej(nk-r~'rr1,)Tl, k:::::::.-k k=...-l< 1<[(\J 5l nl4t [{\J -= r L "Z(bJ e.-l.(lt<- () i t:~q.., f\f'.j.e,-j s:t",il' R \ K::.-R z, / - ~ ~~"'LCK]~--\~K+.Q~)(\ -t ~ b~j~&.q K-JS k:::...,]s 6.02 Spring 2010 Lecture 14, Slide #3

9 Frequency Axes Alternatives Fourier Coefficient versus Radial Frequency Careless about end case Fourier Coefficient versus Coefficient Index Fourier Coefficient versus Sampling Frequency

10 DTFS of x (versus 4Mhz sampling frequency)

11 DTFS of modulation using 400khz and 800khz cosine multiplication

12 Mod with 400khz and 800khz cosines Demod with a 400khz cosine

13 Demodulation using cosine multiplication and a Low-pass Filter (Ideal Channel) Low Pass Filter

14 Mod by multiplication with nearby cosines (400khz, 520khz)

15 Demod of nearby cosine case (at 400khz)

16 Modulation/Demodulation using cosine multiplication and I/O LPF s Low Pass Filter Low Pass Filter x 1 [n] x 1 [n] Low Pass Filter. cos 1 n.. Ideal Channel cos. 1 n.. Low Pass Filter x P [n] x P [n] cos P n cos P n

17 LPF d before cosine multiplication case LPF then Modulation by Cosine Multiplication After Demodulation by Cosine Multiplication

18 Mod by LPF then Sine Multiplication, Demod by Cosine Multiplication then LPF Low Pass Filter Low Pass Filter x 1 [n] x 1 [n] Low Pass Filter... Ideal Channel cos. 1 n.. Low Pass Filter x P [n] x P [n] cos P n

19 DTFS of Mod by sine multiplication

20 DTFS of Demod by cos multiplication, From Mod by sin multiplication

21 Mod by Cos Multiplication with Channel Delay Low Pass Filter Low Pass Filter x 1 [n] x 1 [n] Low Pass Filter. cos 1 n.. D-sample Delay Channel...??? Low Pass Filter x P [n] x P [n] cos P n??? With what shall I demodulation it dear Liza, dear Liza?

22 ( J -." I, }') I \ I, \ I, 1'l t, ~ I.../'. Modv of X [f\-'(). -e..- x C~-

23 I -- \ I -- -, (~ -,A ~ ~vrv\1f\c. X Cf\l is ~ ~ 1\ J LM \~c.rj 'Qr J L.. JL.- ~(~\ X Cf\\.~ II )~ 1.. jll. f) 7\ / '7 (...\J ~..,\(..., _,... Ii"- " t\. 7 i /~ h ~( ;~MjDr~ ~y, /'\ "- Ir( \ ) ("JS..HJI --~ r~ ----~-- ---L - Pt= --'..,---.-.i->: ~..,.---,...f'" "" /' <" J.1'. / i A \ / J~T? 0- ~ "'-~c.....-~ J -., rr-: ~ ;, - J. /\...;- \.r r- I,,' -!'Ul-Vj J \... - J i. ' '\.; ~' ~... I 7\ J.-", j J I 7 />::,. I. - ~\.- I I -~I~ ~ I C- 05.1[.j) - 2-.eli'. (1 A ~ 11\ fi V\ ~ 'I f 'X [~ - 0 "'".. USA.., \1 r.. v _ --..., ~ Ls-, I

24 Example: Consider modulation and demodulation using sine and cosine multiplication. The signal, x[n], is multiplied by a cosine, then transmitted through a channel with a D-sample delay, then multiplie d by a sine and a cosine. For this example, we will assume periodicity, with a N=1000 sample period. Then, the only frequencies that are periodic with period 1000 are multiples of As a specific case, let x[n] be given by and be modulated by. Note that if the channel has a five-sample delay, then

25 The plots on the following slides show the impact of a fivesample channel delay on modulation and demodulation for four different modulation frequencies (but always a fivesample delay). The different modulation frequencies will change the phase shift introduced by the delay, and generate different outputs from sine and cosine demodulation. In particular, the modulation frequencies considered are 25, 50, 75 and 100 times.

26 Plots of x[n] versus n, Real and Imaginary parts of X[k] versus Note: 0.3 for coefficient 0 and ½*0.3 for coefficients -5, -4,-3 and 3, 4, 5

27 Modulation using multiplication by Cos25 Omega_1 n Note: ½*0.3 for coefficient 25 and -25, 1/4*0.3 for coefficients -30, -29,-28, 22,-21, -20 and 30, 29, 28, 22, 21, 20

28 Plots of x after modulation without and with 5 sample delay (note periodicity!)

29 Modulation with Cos25 Omega_1 n multiplication followed by 5 sample delay Note impact of pi/4 phase shift (peaks are 0.3*½ *sqrt(2)/2 for real and imaginary part).

30 Demod by Cos25 Omega_1 n multiplication after delay

31 Demod by Sin 25 Omega_1 n multiplication after delay

32 Modulation by Cos50 Omega_1 n multiplication Note: ½*0.3 for coefficient 50 and -50, 1/4*0.3 for coefficients -55, -54, -53, -47,-46, -45 and 55, 54, 53,47,46, 45

33 Modulation by Cos50 Omega_1 n multiplication followed by 5 sample delay Note impact of pi/2 phase shift (peaks are 0.3*½ but are now in the imaginary part).

34 Demod by Cos50 Omega_1 n multiplication after delay

35 Demod by Sin 50 Omega_1 n multiplication after delay

36 Modulation by Cos75 Omega_1 n multiplication Note: ½*0.3 for coefficient 75 and -75 and 1/4*0.3 for coefficients -80, -79,-78, -72,-71,-70, 80, 79, 78, 72,71, 70

37 Modulation by Cos 75 Omega_1 n multiplication followed by 5 sample delay Note impact of approximately 3pi/4 phase shift (peaks are 0.3*½*sqrt(2)/2 for real and imaginary part, and opposite in sign from 25 Omega_1 case.

38 Demod by Cos75 Omega_1 n multiplication after delay

39 Demod by Sin 75 Omega_1 multiplication after delay

40 Modulation by Cos100 Omega_1 n multiplication Note: ½*0.3 for coefficient 100 and -100, and 1/4*0.3 for coefficients -105, -104, -103, -97, -96, -95, 105, 104, 103, 97, 96, 95

41 Mod by Cos 100 Omega_1 n multiplication followed by 5 sample delay Note impact of pi phase shift (peaks are 0.3* ½ and real, but flipped in sign).

42 Demod by Cos100 Omega_1 n multiplication after delay

43 Demod by Sin 100 Omega_1 n multiplication after delay