I. Previously on IET

1. I. Previously on IET

2. Basic Blocks of Digital Communications Analog-to-Digital Converter Source of continuous-time (i.e., analog) message signal Encoder Band Pass Modulated Signal Quantizer Sampler Low pass Filter Modulation Transmitting Filter m-ary Symbol Encoder

3. TS 0 Square Pulse is a Time-Limited Signal Time-Limited Signal = Frequency Unlimited Spectrum Fourier Transform 0 -3/TS -2/TS -1/TS 1/TS 2/TS 3/TS It is desirable for transmitted signals to be band-limited (limited frequency spectrum) WHY? Guarantee completely orthogonal channels for pass-band signals

4. Inter-symbol Interference (ISI) Frequency Limited Spectrum=Time-Unlimited Signals • A time unlimited signal means inter-symbol interference (ISI) • Neighboring symbols affect the measured value and the corresponding decision at sampling instants Sampling Instants yk(iTS) yk(t)

5. Nyquist Criterion for No ISI • For a given symbol transmitted at iTS yk(t) xk(t) sk(t) yk(TS) Transmitting Filter g(t) Receiving Filter g (TS-t) + Sample at t=TS wk(t) Assume AWGN Noise wk(t) is negligible yk(t) yk(TS) Transmitting Filter g(t) Receiving Filter g (TS-t) Sample at t=TS z(t)=g(t)* g(TS-t)

6. Pulse-shaping with Raised-Cosine Filter z(t): Impulse Response Z(f): Spectrum (Transfer Function) Z(f) T: symbol interval RS: symbol rate r: roll-off factor Raised Cosine Filter Bandwidth = RS(1+r)/2

7. Examples • An analog signal of bandwidth 100 KHz is sampled according to the Nyquist sampling and then quantized and represented by 64 quantization levels. A 4-ary encoder is adopted and a Raised cosine filter is used with roll off factor of 0.5 for base band transmission. Calculate the minimum channel bandwidth to transfer the PCM wave • An analog signal of bandwidth 56 KHz is sampled, quantized and encoded using a quaternary PCM system with raised-cosine spectrum. The rolloff factor is 0.6. If the total available channel bandwidth is 2048 KHz and the channel can support up to 10 users, calculate the number of representation levels of the Quantizer.

8. Phase Shift Keying (PSK) Modulation 1 0 1 1 0 1 Base band Signal X(t) Band Pass Signal Y(t)

9. PSK Demodulation X(t)[2cos2(2πfct)] X(t)cos(2πfct) Low Pass Filter X(t) x 2cos(2πfct) X(t)[2cos2(2πfct)]=X(t)[1+cos(4πfct)] X(t)[2cos2(2πfct)]=X(t) +X(t)cos(4πfct)] Base band Signal (i.e., low frequency content) High frequency content

10. Orthogonality of sin and cos Functions X(t)[2sin(2πfct)cos(2πfct)] X(t)cos(2πfct) Low Pass Filter x 0 2sin(2πfct) X(t)[2sin(2πfct)cos(2πfct)]=X(t) sin(4πfct)] High frequency content

11. Quadrature- PSK Modulation (QPSK) XI(t)cos(2πfct) XI(t) x Y(t) cos(2πfct) + X(t) Serial-to-Parallel XQ(t)sin(2πfct) XQ(t) x sin(2πfct)

12. QPSK Demodulation X (t ) I x Low Pass Filter X(t) Parallel-to-Serial Y(t ) 2cos(2πfct) x Low Pass Filter X (t ) Q 2sin(2πfct)

13. Modulation in Time-Limited Communications Cosine Modulation Binary Encoder Transmitting Filter Binary Symbols In Phase Modulation  Rectangular Filter ES=(1)2×1=1 Time Representation 1 TS 1 Frequency Representation TS f 0 0 f -fc fc Time Representation ES=(-1)2×1 TS -1 0 Frequency Representation -fc fc 0 0 f f -TS

14. Modeling of In phase Modulation Cosine Modulation Binary Encoder Transmitting Filter ES=A2 -A A

15. Modulation in Band-Limited Communications Cosine Modulation Binary Encoder Transmitting Filter Binary Symbols In Phase Modulation  Raised Cosine Filter Time Representation ES=(1)2×1=1 1 t t 1 Frequency Representation 1/RS f f -fc+ RS/2 -RS/2 0 fc+ RS/2 fc- RS/2 -fc- RS/2 RS/2 0 -fc fc Time Representation ES=(-1)2×1 t Bit Rate = RS Bandwidth = RS 1 b/s/Hz t -1 0 Frequency Representation -fc- RS/2 -fc+ RS/2 fc RS/2 -fc fc- RS/2 -RS/2 fc+ RS/2 0 0 f -1/RS 15

16. Modeling of In phase Modulation Cosine Modulation Binary Encoder Transmitting Filter ES=A2 -A A

17. Modulation in Time-Limited Communications Sine Modulation Binary Encoder Transmitting Filter Binary Symbols In Quadrature Modulation  Rectangular Filter ES=(1)2×1=1 Time Representation 1 TS 1 Frequency Representation TS fc f 0 -fc 0 f Time Representation ES=(-1)2×1 TS -1 0 Frequency Representation 0 -fc f 0 f fc -TS 17

18. Modeling of In phase Modulation Sine Modulation Binary Encoder Transmitting Filter ES=A2 jA -jA

19. Modulation in Band-Limited Communications Sine Modulation Binary Encoder Transmitting Filter Binary Symbols In Quadrature Modulation  Raised Cosine Filter Time Representation ES=(1)2×1=1 1 t t 1 Frequency Representation fc 1/RS fc- RS/2 fc+ RS/2 f f -RS/2 0 -fc- RS/2 RS/2 -fc+ RS/2 0 -fc Time Representation ES=(-1)2×1 t Bit Rate = RS Bandwidth = RS 1 b/s/Hz t -1 0 Frequency Representation -fc- RS/2 -fc+ RS/2 RS/2 -fc -RS/2 0 0 f fc- RS/2 fc fc+ RS/2 -1/RS 19

20. Modeling of In phase Modulation Sine Modulation Binary Encoder Transmitting Filter ES=A2 jA -jA

21. Modulation Constellations BPSK QPSK 1 b/s/Hz 2 b/s/Hz 8-QPSK 16 QAM 3 b/s/Hz 4 b/s/Hz

22. Basic Communication Model in AWGN N Detection Performance: • Correct Detection • S = S* • Erroneous Detection • S ≠ S* S R S* Detection TX RX + Channel Model R=S+N

23. BPSK Modulation over AWGN Channels ES  Energy per Symbol

24. BPSK Modulation over AWGN Channels Gaussian Noise 0

25. BPSK Modulation over AWGN Channels Received signal distribution given transmitted 0

26. BPSK Modulation over AWGN Channels Error Calculation given transmitted Symmetry of Gaussian Distribution Let 0

27. BPSK Modulation over AWGN Channels Received signal distribution given transmitted 0

28. BPSK Modulation over AWGN Channels Error Calculation given transmitted Let 0

29. BPSK Modulation over AWGN Channels Signal Power & Symbol Error Performance 0

30. BPSK Modulation over AWGN Channels Signal Power & Symbol Error Performance 0

31. BER of PSK over AWGN Channels Notes: • Define N0 Total Noise Power • N0/2  Noise Power over Cosine axis, i.e., σ2=N0/2 • Each symbol corresponds to a single bit • Eb = ES • Pb = Pe