I. Previously on IET

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

Square Pulse is a Time-Limited Signal Time-Limited Signal = Frequency Unlimited Spectrum Fourier Transform TS -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

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(t) yk(iTS)

Nyquist Criterion for No ISI For a given symbol transmitted at iTS yk(t) sk(t) xk(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)

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

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.

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

PSK Demodulation x X(t)[2cos2(2πfct)] =X(t)[1+cos(4πfct)] Low Pass Filter x X(t) 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

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

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

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

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

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

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

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

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

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

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

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

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

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

BPSK Modulation over AWGN Channels ES  Energy per Symbol

BPSK Modulation over AWGN Channels Gaussian Noise

BPSK Modulation over AWGN Channels Received signal distribution given transmitted

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

BPSK Modulation over AWGN Channels Received signal distribution given transmitted

BPSK Modulation over AWGN Channels Error Calculation given transmitted Let

BPSK Modulation over AWGN Channels Signal Power & Symbol Error Performance

BPSK Modulation over AWGN Channels Signal Power & Symbol Error Performance

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