Digital Communication Transmitter

Introduction to Digital „ A simplified block diagram: Modulation antenna

waveforms bits (symbols) 0110 interleaver User RF Instructor: M.A. Ingram Interface Encoder Modulator Front more bits End ECE4823 011001 Frequency up-conversion and amplification

Digital Communications Receiver Symbols

„ A simplified block diagram: „ In each symbol period, Ts, a digital antenna waveforms modulator maps N coded bits word to a (faded symbols transmitted waveform from a set of plus noise) bits deinterleaver 0110 N RF User M=2 possible waveforms Front Demodulator Decoder Interface End „ Each waveform corresponds to an “soft detections” Low-noise amplification of coded bits information symbol, x and frequency 011001 n down-conversion „ For Binary symbols, N=1

Detection Definitions

„ The job of the receiver is to determine „ Bit Rate (bits per sec or bps)

which symbols were sent and to R = N / TS reconstruct the bit stream that created „ Bandwidth Efficiency (bps/Hz) them η B = R / B where B is the bandwidth occupied by the signal

1 Shannon Theorem Pulses

„ In a non-fading channel, the maximum „ A symbol period, Ts, suggests a bandwidth efficiency, or Shannon localization in time Capacity is „ Localization in frequency is also necessary to enable frequency division η = log ()1+ SNR BMAX 2 multiplexing „ Regulatory agencies provide spectral SNR = signal-to-noise ratio masks to limit the distribution of power in the frequency domain

Example: Linear Modulation Rectangular Pulses

„ s(t) is the output of the modulator „ Suppose p(t) is a rectangular pulse

„ g(t) is the complex envelope „ This pulse is not used in practice, but is OK „ p(t) is the basic pulse for illustration

„ x is the nth symbol n p(t) „ A is the amplification in the transmitter 2 j 2πfct s(t) = Re{g(t)e } TS g(t) = A x p(t − nT ) t ∑ n S 0 T n S

Binary Phase Shift Keying (BPSK) BPSK-Modulated Carrier

„ For BPSK, each symbol carries one bit „ The information is in the phase of the

of information xn ∈{}−1,1 carrier g(t) s(t)

2 1 0 0 1 0 1 1 2 1 0 0 1 0 1 1 A A TS TS 2T S L t L t

0 TS 2 2 − A − A TS TS

2 Power Spectral Density (PSD) of Linear Modulation Bandwidth Properties

„ Assume that the symbol sequence {xn} is iid „ The RF bandwidth of the modulated and zero mean. Then, the PSD of s(t) is carrier is two times the baseband 1 bandwidth of S ( f ) = []S ( f − f ) + S (− f − f ) s 2 g c g c 2 2 A E{ xn } 2 where S g ( f ) = P( f ) 2 2 2TS A E{ xn } 2 S g ( f ) = P( f ) 2TS which is clearly seen to depend on the and P ( f ) is the Fourier Transform of p (t) bandwidth of the pulse

Fourier Transform of the PSD for BPSK and the Rectangular Pulse Rectangular Pulse

2 2 „ S s ( f ) P( f ) = 2TS sinc ( fTS ) 2 +∞ TS „ | 2 xn| =1 P( f ) = p(t) exp − j2πft dt = exp − j2πft dt 2 ∫ ()∫ () 2 T A E{ xn } 2 −∞ 0 S − fc fc S g ( f ) = P( f ) T S 2TS 2 exp()− j2πft 2 exp()− j2πfTS −1 = = 2 2 = A sinc ()fTS TS − j2πf 0 TS − j2πf

2 exp()− jπfTS 2 A = sin()πfT = exp (− jπfT )2T sinc()fT 2 2 S S S S S s ( f ) = []sinc ()[]f − fc TS + sinc ()[]− f − fc TS πf TS 2

Received Signal BPSK Demodulator

„ The received signal has been attenuated by path loss „ The output of the correlator is a sequence of and multipath fading and has added noise noisy versions of the transmitted symbol sequence r(t) = a(t)A x p(t − nT ) cos(2πf t +θ ) + n(t) demodulated ∑ n S c c received symbols n nTS .95,-1.03,-1.02,.93,-.85,.81,.74 r(t) ()• dt 2 ∫()n−1 T a(t)A 1 0 0 1 0 1 1 S To deinterleaver and decoder TS Channel Gain L t 1 2 cos()2πfct Aa(t) TS

Automatic Gain Control

3 Integrate and Dump Output Conditional PDFs

„ Consider one symbol period and the „ Recall R = xn εb + vn rectangular pulse: channel gains and TX power lumped into εb

TS TS   2  2ε  2 2    b    2 R = r(t) cos()2πfct dt = xn cos()2πfct + n(t) cos()2πfct dt 1 ()r + εb  r −  ∫ ∫   f (r) = exp−  1 ()εb 0 TS 0 TS TS R|xn =−1 f (r) = exp−     N  R|xn =1  N0   0  N   N0   2π 2   0  2 TS TS     2π     x 2  2    2     2   = n 1 dt + n(t)cos 2πf t dt = x + v  2    εb ∫ ∫ ()c n εb n TS 0 TS 0 r „ v is a zero-mean Gaussian RV with variance n − εb εb N σ 2 = 0 v 2

Conditional Probability of Bit Conditional Probability of Error Error Expression

„ If xn is -1, then an error happens if R>0     +∞ 0 − − ε  2   ()b   εb  f R|x =1(r | xn = −1) f R|x =−1(r | xn = −1)dr = Q = Q n ∫ n  N   N  0  0   0   2  r

− εb εb „ Same for other kind of error, by symmetry

P(error | xn = −1) = P(R > 0 | xn = −1) +∞ = f (r | x = −1)dr ∫ R|xn =−1 n 0

Unconditional Probability of Bit How to Improve System Error Performance

„ Assume the two possible values of xn „ Increase symbol energy εb

are equally likely „ Decrease average noise power N0 / 2

1 1 P(error) = P(error | x = −1) + P(error | x = 1) 2 n 2 n  2ε  P(error) = Q b   2   N   εb   0  = Q   N0 

4 References

„ [Rapp, ’02] T.S. Rappaport, Wireless Communications, Prentice Hall, 2002

„ [Stuber, ’01] Gordon Stuber, Principles of Mobile Communication, 2nd ed, Kluwer Academic Publishers, 2001.

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