Baseband Demodulation-Digital Communication Systems-Lecture Slides, Slides of Digital Communication Systems

Download Baseband Demodulation-Digital Communication Systems-Lecture Slides and more Slides Digital Communication Systems in PDF only on Docsity! 1 1 Digital Communication Systems Dr. Shurjeel Wyne Lecture 5 Baseband Demodulation and Detection 2 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 2 3 Bit stream (Data bits) Formatting and transmission of baseband signal Information (data) rate: Symbol rate : For real time transmission: Sampling at rate (sampling time=Ts) Quantizing each sampled value to one of the L levels in quantizer. Encoding each quant. value to bits Encode Pulse modulateSample Quantize Pulse waveforms (baseband signals)Format Digital info. Textual info. Analog info. Mapping every data bits to a symbol out of M symbols and transmitting a baseband waveform with duration T ss Tf /1= Ll 2log= Mk 2log= 4 Today we are going to talk about: Receiver structure Demodulation Detection Impact of AWGN and ISI on Transmitted Signal Steps for receiver design Matched filter receiver Correlator receiver Representation of signals as vectors (signal space) an important tool for understanding: Modulation schemes Detection operation 5 9 Combined effect of AWGN and ISI )()()()( tnthtstr ci +∗= )75.0(5.0)()( Tttthc −−= δδ 10 Receiver Tasks Demodulation Waveform recovery and preparing the received signal for detection : Improving the ratio of signal power to the noise power (SNR) using a matched filter Reducing ISI using equalizer (remove channel distortion) Sampling the recovered waveform Detection Estimate the transmitted symbol based on the received sample 6 11 Receiver structure Frequency down-conversion Receiving filter Equalizing filter Threshold comparison For bandpass signals Compensation for channel induced ISI Baseband pulse (possibly distorted) Sample(test statistic) Baseband pulseReceived waveform Step 1 – waveform to sample transformation Step 2 – decision making )(tr )(Tz im̂ Demodulate & Sample Detect 12 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. Baseband vs Bandpass Detection at the receiver 7 13 Steps in designing the receiver Model the received signal Find optimum solution for receiver design with following goals: 1. Maximize SNR 2. Minimize ISI Find separate solutions for each of the above goals. 14 Simplified model: Received signal in AWGN )(thc)(tsi )(tn )(tr )(tn )(tr)(tsi Ideal channels )()( tthc δ= AWGN AWGN )()()()( tnthtstr ci +∗= )()()( tntstr i += Model of received signal 10 19 Implementation of matched filter receiver ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Mz z M 1 z= )(tr )(1 Tz )(*1 tTs − )(* tTsM − )(TzM z Bank of M matched filters Matched filter output: Observation vector )()( tTstrz ii −∗= ∗ Mi ,...,1= ),...,,())(),...,(),(( 2121 MM zzzTzTzTz ==z 20 Example of implementation of matched filter receivers ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 2 1 z z z= )(tr )(1 Tz )(2 Tz z Bank of 2 matched filters T t )(1 ts T t )(2 ts T T0 0 T A T A− T A− T A 0 0 11 21 Correlator receiver The matched filter output at the sampling time, can also be realized as the correlator output. Matched filtering, i.e. convolution with si*(T-t) can be alternatively viewed as integration with si*(t), i.e. correlation >=<= ∗= ∫ )(),()()( )()()( * 0 tstrdsr TrThTz i T opt τττ 22 Implementation of correlator receiver dttstrz i T i )()( 0 ∫= ∫ T 0 )(1 ts∗ ∫ T 0 )(ts M∗ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Mz z M 1 z= )(tr )(1 Tz )(TzM z Bank of M correlators Correlators output: Observation vector ),...,,())(),...,(),(( 2121 MM zzzTzTzTz ==z Mi ,...,1= 12 23 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -1 -0.5 0 0.5 1 V ol ta ge in pu t Time [s], at filter input Symbol interval of width 0.5 s, consisting of 5 cycles of a sinusoid 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 0 1 2 N or m al iz ed V ol ta ge o ut pu t Time [s], at filter output Comparison of outputs from MF and Correlator MF output Correlator output Sympol Interval T=0.5 sec 24 Signal space What is a signal space? An N-dimensional orthogonal vector space that is used to represent modulation signals/waveforms as vectors. Why do we need a signal space? It is a means to convert signals to vectors and vice versa. It is a means to calculate signal energy and Euclidean distances between signals. Why are we interested in Euclidean distances between signals? For detection purposes: The received signal is transformed to a received vector. The signal alternative that has the minimum distance to the received signal is estimated as the transmitted signal. 15 29 Example of distances in signal space )(1 tψ )(2 tψ ),( 12111 aa=s ),( 22212 aa=s ),( 32313 aa=s ),( 21 zz=z zsd ,1 zsd ,2zs d ,3 The Euclidean distance between signals z(t) and s(t): 3,2,1 )()()()( 222 2 11, = −+−=−= i zazatztsd iiizsi 1E 3E 2E 30 Example of an orthonormal basis functions Example: 2-dimensional orthonormal signal space Example: 1-dimensional orthonormal signal space 1)()( 0)()()(),( 0)/2sin(2)( 0)/2cos(2)( 21 2 0 121 2 1 == =>=< ⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧ <≤= <≤= ∫ tt dttttt TtTt T t TtTt T t T ψψ ψψψψ πψ πψ T t )(1 tψ T 1 0 )(1 tψ )(2 tψ 0 1)(1 =tψ )(1 tψ0 16 31 Signal space – cont’d Any arbitrary finite set of waveforms where each member of the set is of duration T, can be expressed as a linear combination of N orthogonal waveforms where . where { }Mii ts 1)( = { }N jj t 1 )( = ψ MN ≤ ∑ = = N j jiji tats 1 )()( ψ Mi ,...,1= MN ≤ dttts K tts K a T ji j ji j ij )()( 1)(),(1 0 *∫>=<= ψψ Tt ≤≤0Mi ,...,1= Nj ,...,1= ),...,,( 21 iNiii aaa=s 2 1 ij N j ji aKE ∑ = = Vector representation of waveform Waveform energy 32 Signal space - cont’d ∑ = = N j jiji tats 1 )()( ψ ),...,,( 21 iNiii aaa=s ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ iN i a a M 1 )(1 tψ )(tNψ 1ia iNa )(tsi ∫ T 0 )(1 tψ ∫ T 0 )(tNψ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ iN i a a M 1 ms= )(tsi 1ia iNa ms Waveform to vector conversion Vector to waveform conversion i i 17 33 Example of projecting signals to an orthonormal signal space ),()()()( ),()()()( ),()()()( 323132321313 222122221212 121112121111 aatatats aatatats aatatats =⇔+= =⇔+= =⇔+= s s s ψψ ψψ ψψ )(1 tψ )(2 tψ ),( 12111 aa=s ),( 22212 aa=s ),( 32313 aa=s Transmitted signal alternatives dtttsa T jiij )()( 0 ∫= ψ Tt ≤≤0Mi ,...,1=Nj ,...,1= 34 Signal space – cont’d To find orthonormal basis functions for a given set of signals, Gram-Schmidt procedure can be used. Gram-Schmidt procedure: Given a signal set , compute an orthonormal basis 1. Define 2. For compute If let If , do not assign any basis function. 3. Renumber the basis functions such that basis is This is only necessary if for any i in step 2. Note that { }Mii ts 1)( = { }Njj t 1)( =ψ )(/)(/)()( 11111 tstsEtst ==ψ Mi ,...,2= ∑ − = ><−= 1 1 )()(),()()( i j jjiii tttststd ψψ 0)( ≠tdi )(/)()( tdtdt iii =ψ 0)( =tdi { })(),...,(),( 21 ttt Nψψψ 0)( =tdi MN ≤