Digital Communications I: Modulation and Coding Course

Digital communications I: Modulation and Coding Course Period 3 - 2007 Catharina Logothetis Lecture 6 Lecture 6 2 Last time we talked about:  Signal detection in AWGN channels  Minimum distance detector  Maximum likelihood  Average probability of symbol error  Union bound on error probability  Upper bound on error probability based on the minimum distance Lecture 6 3 Today we are going to talk about:  Another source of error:  Inter-symbol interference (ISI)  Nyquist theorem  The techniques to reduce ISI  Pulse shaping  Equalization Lecture 6 4 Inter-Symbol Interference (ISI)  ISI in the detection process due to the filtering effects of the system  Overall equivalent system transfer function  creates echoes and hence time dispersion  causes ISI at sampling time )()()()( fHfHfHfH rct = i ki ikkk snsz ∑ ≠ ++= α Lecture 6 5 Inter-symbol interference  Baseband system model  Equivalent model Tx filter Channel )(tn )(tr Rx. filter Detector k z kTt = { } k x ˆ {} k x 1 x 2 x 3 x T T )( )( fH th t t )( )( fH th r r )( )( fH th c c Equivalent system )( ˆ tn )( tz Detector k z kTt = { } k x ˆ {} k x 1 x 2 x 3 x T T )( )( fH th filtered noise )()()()( fHfHfHfH rct = Lecture 6 6 Nyquist bandwidth constraint  Nyquist bandwidth constraint:  The theoretical minimum required system bandwidth to detect Rs [symbols/s] without ISI is Rs/2 [Hz].  Equivalently, a system with bandwidth W=1/2T=Rs/2 [Hz] can support a maximum transmission rate of 2W=1/T=Rs [symbols/s] without ISI.  Bandwidth efficiency, R/W [bits/s/Hz] :  An important measure in DCs representing data throughput per hertz of bandwidth.  Showing how efficiently the bandwidth resources are used by signaling techniques. Hz][symbol/s/ 2 22 1 ≥⇒≤= W R W R T ss Lecture 6 7 Ideal Nyquist pulse (filter) T2 1 T2 1− T )( fH f t )/sinc()( Ttth = 1 0 T T2 T− T2− 0 T W 2 1 = Ideal Nyquist filter Ideal Nyquist pulse Lecture 6 8 Nyquist pulses (filters)  Nyquist pulses (filters):  Pulses (filters) which results in no ISI at the sampling time .  Nyquist filter:  Its transfer function in frequency domain is obtained by convolving a rectangular function with any real even-symmetric frequency function  Nyquist pulse:  Its shape can be represented by a sinc(t/T) function multiply by another time function.  Example of Nyquist filters: Raised-Cosine filter Lecture 6 9 Pulse shaping to reduce ISI  Goals and trade-off in pulse-shaping  Reduce ISI  Efficient bandwidth utilization  Robustness to timing error (small side lobes) Lecture 6 10 The raised cosine filter  Raised-Cosine Filter  A Nyquist pulse (No ISI at the sampling time) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ > <<− ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − −+ −< = Wf WfWW WW WWf WWf fH ||for 0 ||2for 2|| 4 cos 2||for 1 )( 0 0 0 2 0 π Excess bandwidth: 0 WW − Roll-off factor 0 0 W WW r − = 10 ≤≤ r 2 0 0 00 ])(4[1 ])(2cos[ ))2(sinc(2)( tWW tWW tWWth −− − = π [...]... with ISI: Binary-PAM, SRRQ pulse Non-ideal channel and no noise hc (t ) = δ (t ) + 0.7δ (t − T ) Lecture 6 23 Example of eye pattern with ISI: Binary-PAM, SRRQ pulse … AWGN (Eb/N0=20 dB) and ISI hc (t ) = δ (t ) + 0.7δ (t − T ) Lecture 6 24 Example of eye pattern with ISI: Binary-PAM, SRRQ pulse … AWGN (Eb/N0=10 dB) and ISI hc (t ) = δ (t ) + 0.7δ (t − T ) Lecture 6 25 Equalizing filters … Baseband system... channel (no noise and no ISI) Lecture 6 16 Example of eye pattern: Binary-PAM, SRRQ pulse … AWGN (Eb/N0=20 dB) and no ISI Lecture 6 17 Example of eye pattern: Binary-PAM, SRRQ pulse … AWGN (Eb/N0=10 dB) and no ISI Lecture 6 18 Equalization – cont’d Step 1 – waveform to sample transformation Step 2 – decision making Demodulate & Sample Detect z (T ) r (t ) Frequency down-conversion For bandpass signals... Receiving filter Equalizing filter Threshold comparison Compensation for channel induced ISI Baseband pulse (possibly distored) Lecture 6 Baseband pulse Sample (test statistic) 19 ˆ mi Equalization ISI due to filtering effect of the communications channel (e.g wireless channels) Channels behave like band-limited filters Hc ( f ) = Hc ( f ) e jθ c ( f ) Non-constant amplitude Non-linear phase Amplitude... 5 0.5 r =1 0.5 r =1 r = 0.5 r =0 −1 − 3 −1 T 4T 2T 0 1 3 2T 4T 1 T Rs Baseband W sSB= (1 + r ) 2 Lecture 6 − 3T − 2T − T 0 T 2T Passband W DSB= (1 + r ) Rs 11 3T Pulse shaping and equalization to remove ISI No ISI at the sampling time H RC ( f ) = H t ( f ) H c ( f ) H r ( f ) H e ( f ) Square-Root Raised Cosine (SRRC) filter and Equalizer H RC ( f ) = H t ( f ) H r ( f ) H r ( f ) = H t ( f ) = H RC... pulse shaping Amp [V] Baseband tr Waveform Third pulse t/T First pulse Second pulse Data symbol Lecture 6 13 Example of pulse shaping … Raised Cosine pulse at the output of matched filter Amp [V] Baseband received waveform at the matched filter output (zero ISI) t/T Lecture 6 14 Eye pattern Eye pattern:Display on an oscilloscope which sweeps the system response to a baseband signal at the rate 1/T... output is forced to be zero at N sample points on each side: k =0 ⎧1 z (k ) = ⎨ ⎩0 k = ±1, ,± N Adjust N {cn }n=− N Mean Square Error (MSE) equalizer: The filter taps are adjusted such that the MSE of ISI and noise power at the equalizer output is minimized Adjust {c n }nN= − N [ min E ( z (kT ) − ak ) 2 Lecture 6 ] 29 Example of equalizer 2-PAM with SRRQ Non-ideal channel hc (t ) = δ (t ) + 0.3δ (t − T . Digital communications I: Modulation and Coding Course Period 3 - 2007 Catharina Logothetis Lecture. rct = Lecture 6 6 Nyquist bandwidth constraint  Nyquist bandwidth constraint:  The theoretical minimum required system bandwidth to detect Rs [symbols/s]