Digital Communication I: Modulation and Coding Course Period - 2007 Catharina Logothetis Lecture Last time we talked about: Transforming the information source to a form compatible with a digital system Sampling Aliasing Quantization Uniform and non-uniform Baseband modulation Binary pulse modulation M-ary pulse modulation M-PAM (M-ay Pulse amplitude modulation) Lecture Formatting and transmission of baseband signal Digital info Format Textual source info Analog info Bit stream (Data bits) Sample Sampling at rate f s = / Ts (sampling time=Ts) Quantize Pulse waveforms (baseband signals) Pulse modulate Encode Encoding each q value to l = log L bits (Data bit duration Tb=Ts/l) Mapping every m = log M data bits to a symbol out of M symbols and transmitting a baseband waveform with duration T Quantizing each sampled value to one of the L levels in quantizer Information (data) rate: Rb = / Tb [bits/sec] Symbol rate : R = / T [symbols/sec] For real time transmission: Rb = mR Lecture 3 Quantization example amplitude x(t) 111 3.1867 Quant levels 110 2.2762 101 1.3657 100 0.4552 boundaries 011 -0.4552 010 -1.3657 001 -2.2762 xq(nTs): quantized values x(nTs): sampled values 000 -3.1867 Ts: sampling time PCM codeword t 110 110 111 110 100 010 011 100 100 011 PCM sequence Lecture Example of M-ary PAM Assuming real time tr and equal energy per tr data bit for binary-PAM and 4-ary PAM: • 4-ary: T=2Tb and Binay: T=Tb A2 = 10B • Binary PAM (rectangular pulse) 4-ary PAM (rectangular pulse) 3B A ‘1’ B T ‘10’ ‘0’ -A T T T -B ‘00’ ‘11’ ‘01’ T T -3B Lecture Example of M-ary PAM … Ts 2Ts 2.2762 V Tb 1.3657 V 2Tb 3Tb 4Tb 5Tb 6Tb 1 Rb=1/Tb=3/Ts R=1/T=1/Tb=3/Ts T 2T 3T 4T 5T 6T Rb=1/Tb=3/Ts R=1/T=1/2Tb=3/2Ts=1.5/Ts T 2T 3T Lecture Today we are going to talk about: Receiver structure Demodulation (and sampling) Detection First step for designing the receiver Matched filter receiver Correlator receiver Lecture Demodulation and detection g i (t ) Bandpass si (t ) Pulse modulate modulate channel transmitted symbol hc (t ) Format mi estimated symbol Format ˆ mi Detect M-ary modulation i = 1, K, M n(t ) Demod z (T ) & sample r (t ) Major sources of errors: Thermal noise (AWGN) disturbs the signal in an additive fashion (Additive) has flat spectral density for all frequencies of interest (White) is modeled by Gaussian random process (Gaussian Noise) Inter-Symbol Interference (ISI) Due to the filtering effect of transmitter, channel and receiver, symbols are “smeared” Lecture Example: Impact of the channel Lecture Example: Channel impact … hc (t ) = δ (t ) − 0.5δ (t − 0.75T ) Lecture 10 Receiver job Demodulation and sampling: Waveform recovery and preparing the received signal for detection: Improving the signal power to the noise power (SNR) using matched filter Reducing ISI using equalizer Sampling the recovered waveform Detection: Estimate the transmitted symbol based on the received sample Lecture 11 Receiver structure Step – waveform to sample transformation Step – decision making Demodulate & Sample Detect z (T ) r (t ) Frequency down-conversion For bandpass signals Received waveform Receiving filter Equalizing filter Threshold comparison Compensation for channel induced ISI Baseband pulse (possibly distored) Lecture Baseband pulse Sample (test statistic) 12 ˆ mi Baseband and bandpass Bandpass model of detection process is equivalent to baseband model because: The received bandpass waveform is first transformed to a baseband waveform Equivalence theorem: Performing bandpass linear signal processing followed by heterodying the signal to the baseband, yields the same results as heterodying the bandpass signal to the baseband , followed by a baseband linear signal processing Lecture 13 Steps in designing the receiver Find optimum solution for receiver design with the following goals: Maximize SNR Minimize ISI Steps in design: Model the received signal Find separate solutions for each of the goals First, we focus on designing a receiver which maximizes the SNR Lecture 14 Design the receiver filter to maximize the SNR Model the received signal si (t ) r (t ) = si (t ) ∗h c (t ) + n(t ) r (t ) hc (t ) n(t ) AWGN Simplify the model: Received signal in AWGN Ideal channels hc (t ) = δ (t ) r (t ) si (t ) r (t ) = si (t ) + n(t ) n(t ) AWGN Lecture 15 Matched filter receiver Problem: Design the receiver filter h(t ) such that the SNR is maximized at the sampling time when si (t ), i = 1, , M is transmitted Solution: The optimum filter, is the Matched filter, given by * h ( t ) = hopt ( t ) = s i (T − t ) * H ( f ) = H opt ( f ) = S i ( f ) exp( − j 2π fT ) which is the time-reversed and delayed version of the conjugate of the transmitted signal h(t ) = hopt (t ) si (t ) T t Lecture T t 16 Example of matched filter y (t ) = si (t ) ∗h opt (t ) h opt (t ) si (t ) A T A2 A T T t T si (t ) 2T t T 3T/2 2T t T y (t ) = si (t ) ∗h opt (t ) h opt (t ) A T t A T T/2 T −A T t A2 T/2 T T/2 t − −A T Lecture A2 17 Properties of the matched filter The Fourier transform of a matched filter output with the matched signal as input is, except for a time delay factor, proportional to the ESD of the input signal Z ( f ) =| S ( f ) |2 exp(− j 2πfT ) The output signal of a matched filter is proportional to a shifted version of the autocorrelation function of the input signal to which the filter is matched z (t ) = Rs (t − T ) ⇒ z (T ) = Rs (0) = Es The output SNR of a matched filter depends only on the ratio of the signal energy to the PSD of the white noise at the filter input Es ⎛S⎞ max⎜ ⎟ = ⎝ N ⎠T N / Two matching conditions in the matched-filtering operation: spectral phase matching that gives the desired output peak at time T spectral amplitude matching that gives optimum SNR to the peak value Lecture 18 Correlator receiver The matched filter output at the sampling time, can be realized as the correlator output z (T ) = hopt (T ) ∗ r (T ) T = ∫ r (τ )si (τ )dτ =< r (t ), s (t ) > * Lecture 19 Implementation of matched filter receiver Bank of M matched filters s (T − t ) * z1 (T ) r (t ) sM (T − t ) * zM ⎡ z1 ⎤ ⎢ M ⎥=z ⎢ ⎥ ⎢zM ⎥ ⎣ ⎦ (T ) Matched filter output: z Observation vector zi = r (t ) ∗ s ∗i (T − t ) i = 1, , M z = ( z1 (T ), z (T ), , z M (T )) = ( z1 , z , , z M ) Lecture 20 Implementation of correlator receiver Bank of M correlators s ∗1 (t ) ∫ T z1 (T ) r (t ) s ∗ M (t ) ∫ T ⎡ z1 ⎤ ⎢M⎥ ⎢ ⎥ ⎢zM ⎥ ⎣ ⎦ =z Correlators output: z Observation vector z M (T ) z = ( z1 (T ), z (T ), , z M (T )) = ( z1 , z , , z M ) T zi = ∫ r (t )si (t )dt i = 1, , M Lecture 21 Implementation example of matched filter receivers s1 (t ) Bank of matched filters A T T z1 (T ) A T t r (t ) T s2 (t ) −A T T t T ⎡z1 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢z2 ⎥ ⎣ ⎦ =z z z (T ) −A T Lecture 22 ... Tb 1 .36 57 V 2Tb 3Tb 4Tb 5Tb 6Tb 1 Rb=1/Tb =3/ Ts R=1/T=1/Tb =3/ Ts T 2T 3T 4T 5T 6T Rb=1/Tb =3/ Ts R=1/T=1/2Tb =3/ 2Ts=1.5/Ts T 2T 3T Lecture Today we are going to talk about: Receiver structure Demodulation... Quantization Uniform and non-uniform Baseband modulation Binary pulse modulation M-ary pulse modulation M-PAM (M-ay Pulse amplitude modulation) Lecture Formatting and transmission of baseband signal Digital... induced ISI Baseband pulse (possibly distored) Lecture Baseband pulse Sample (test statistic) 12 ˆ mi Baseband and bandpass Bandpass model of detection process is equivalent to baseband model because: