Digital Communications I: Modulation and Coding Course Period 3 - 2007 Catharina Logothetis Lecture 4 Lecture 4 2 Last time we talked about:  Receiver structure  Impact of AWGN and ISI on the transmitted signal  Optimum filter to maximize SNR  Matched filter receiver and Correlator receiver Lecture 4 3 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 4 4 Receiver structure Frequency down-conversion Receiving filter Equalizing filter Threshold comparison For bandpass signals Compensation for channel induced ISI Baseband pulse (possibly distored) Sample (test statistic) Baseband pulse Received waveform Step 1 – waveform to sample transformation Step 2 – decision making )(tr )(Tz i m ˆ Demodulate & Sample Detect Lecture 4 5 Implementation of matched filter receiver ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ M z z M 1 z= )(tr )( 1 Tz )( * 1 tTs − )( * tTs M − )(Tz M z Matched filter output: Observation vector Bank of M matched filters )()( tTstrz i i −∗= ∗ Mi , ,1 = ), ,,())(), ,(),(( 2121 MM zzzTzTzTz = =z Lecture 4 6 Implementation of correlator receiver dttstrz i T i )()( 0 ∫ = ∫ T 0 )( 1 ts ∗ ∫ T 0 )(ts M ∗ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ M z z M 1 z= )(tr )( 1 Tz )(Tz M z Correlators output: Observation vector Bank of M correlators ), ,,())(), ,(),(( 2121 MM zzzTzTzTz = =z Mi , ,1 = Lecture 4 7 Today, we are going to talk about:  Detection:  Estimate the transmitted symbol based on the received sample  Signal space used for detection  Orthogonal N-dimensional space  Signal to waveform transformation and vice versa Lecture 4 8 Signal space  What is a signal space?  Vector representations of signals in an N-dimensional orthogonal space  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 signals 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 vectors. The signal which has the minimum distance to the received signal is estimated as the transmitted signal. Lecture 4 9 Schematic example of a signal space ),()()()( ),()()()( ),()()()( ),()()()( 212211 323132321313 222122221212 121112121111 zztztztz aatatats aatatats aatatats =⇔+= =⇔+= =⇔+= = ⇔ + = z s s s ψψ ψψ ψψ ψ ψ )( 1 t ψ )( 2 t ψ ),( 12111 aa = s ),( 22212 aa = s ),( 32313 aa=s ),( 21 zz = z Transmitted signal alternatives Received signal at matched filter output Lecture 4 10 Signal space  To form a signal space, first we need to know the inner product between two signals (functions):  Inner (scalar) product:  Properties of inner product: ∫ ∞ ∞− >=< dttytxtytx )()()(),( * = cross-correlation between x(t) and y(t) > < >=< )(),()(),( tytxatytax ><>=< )(),()(),( * tytxataytx > < + >>=<+< )(),()(),()(),()( tztytztxtztytx [...]... as the Euclidean distance between two signals Lecture 4 11 Example of distances in signal space ψ 2 (t ) s1 = (a11 , a12 ) E1 d s1 , z ψ 1 (t ) z = ( z1 , z 2 ) E3 s 3 = (a31 , a32 ) d s3 , z E2 d s2 , z s 2 = (a21 , a22 ) The Euclidean distance between signals z(t) and s(t): d si , z = si (t ) − z (t ) = (ai1 − z1 ) 2 + (ai 2 − z 2 ) 2 i = 1,2,3 Lecture 4 12 Orthogonal signal space N-dimensional orthogonal... orthonormal Lecture 4 13 Example of an orthonormal bases Example: 2-dimensional orthonormal signal space ⎧ 2 ψ 1 (t ) = cos(2πt / T ) 0≤t ~ < n (t ),ψ (t ) >= 0 j j = 1, , N j = 1, , N Lecture 4 ˆ n(t ) n = (n1 , n2 , , nN ) {n } N independent zero-mean Gaussain random variables with variance var(n j ) = N 0 / 2 j j =1 23 ... 1 (t ) + a12ψ 2 (t ) ⇔ s1 = (a11 , a12 ) s2 (t ) = a21ψ 1 (t ) + a22ψ 2 (t ) ⇔ s 2 = (a21 , a22 ) s3 (t ) = a31ψ 1 (t ) + a32ψ 2 (t ) ⇔ s 3 = (a31 , a32 ) T aij = ∫ si (t )ψ j (t )dt 0 j = 1, , N Lecture 4 i = 1, , M 0≤t ≤T 17 Signal space – cont’d To find an orthonormal basis functions for a given set of signals, Gram-Schmidt procedure can be used Gram-Schmidt procedure: N M ψ Given a signal set {si . Digital Communications I: Modulation and Coding Course Period 3 - 2007 Catharina Logothetis Lecture 4 Lecture 4 2 Last time we talked about: . AWGN and ISI on the transmitted signal  Optimum filter to maximize SNR  Matched filter receiver and Correlator receiver Lecture 4 3 Receiver job  Demodulation