Eecs 455 univ of michigan fall 2012 september 7 2012

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Unformatted text preview: ms Matched Filters In a digital communication system it is usual for the received filter to be matched to the transmitted signal. In this case, if s(t ) is the transmitted signal and is of duration T beginning at 0, we sample the filter output at time T and h(t ) = s(T − t ). This is called the matched filter. The filter output is y (t ) = ∞ −∞ t h(t − α)s(α)d α = t −T t = t −T t = t −T h(t − α)s(α)d α s(T − (t − α))s(α)d α s(α − (t − T ))s(α)d α (This is the autocorrelation of the signal s(t )). EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 61 / 174 Lecture Notes 2 Linear Systems Matched Filter The desired signal is the output sampled at time T . T T y (T ) = 0 h(T − α)s(α)d α = s(α)s(α)d α 0 T = s2 (α)d α 0 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 62 / 174 Lecture Notes 2 Linear Systems Multiuser System In a spread-spectrum system the signal has the form s(t ) = N −1 l =0 al ψ (t − lTc ) so the impulse response of the matched filter has the following form h(t ) = s(T − t ) = N −1 l =0 al ψ (T − t − lTc ) where N is the number of “chips” per bit, 1/Tc is the chip rate, and NTc = T is the data bit duration or the inverse data rate. In this case the implementation of the matched filter can be simplified as follows. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 63 / 174 Lecture Notes 2 Linear Systems Multiuser System Let s(t ) be the filter input then y (t ) = ∞ −∞ = h(t − α)s(α)d α ∞ N −1 −∞ l =1 = N −1 l =0 EECS 455 (Univ. of Michigan) al al ψ (T − t + α − lTc )s(α)d α ∞ −∞ ψ (T − t + α − lTc )s(α)d α. Fall 2012 September 7, 2012 64 / 174 Lecture Notes 2 Linear Systems Matched Filter Let x (t ) be the output of a filter with impulse response ψ (Tc − t ) then x (t ) = ∞ −∞ ψ (Tc − t + α)s(α)d α. Now it is clear that y (t ) = N −1 l =0 EECS 455 (Univ. of Michigan) al x (t − (N − 1 − l )Tc ). Fall 2012 September 7, 2012 65 / 174 Lecture Notes 2 Linear Systems Matched Filter Thus the matched filter can be implemented as a filter matched to the chip waveform followed by a weighted sum. Since we are interested in the sample only at time t = mT we only need the samples of the chip matched filter at multiples of Tc . For example at time t = T the output is N −1 al x ((l + 1)Tc ) y (T ) = l =0 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 66 / 174 Lecture Notes 2 Linear Systems Matched Filter At time t = mT the output is y (mT ) = N −1 l =0 al x ((m − 1)T + (l + 1)Tc ) mN y (mT ) = j =(m−1)N +1 aj −1−(m−1)N x (jTc ) If the spreading sequence is periodic with period N so that aj +N = aj then mN y (mT ) = j =(m−1)N +1 EECS 455 (Univ. of Michigan) Fall 2012 aj −1 x (jTc ). September 7, 2012 67 / 174 Lecture Notes 2 Linear Systems Matched Filter t = jTc ψ (t ) × mN j =(m−1)N +1 y (MT ) x (jTc ) a j −1 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 68 / 174 Lecture Notes 2 Linear Systems Matched Filtering s(t ) +1 t -1 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 69 / 174 Lecture Notes 2 Linear Systems Matched Filtering h(t ) +1 t -1 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 70 / 174 Lecture Notes 2 Linear Systems Matched Filtering 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -1 -2 -3 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 71 / 174 Lecture Notes 2 Linear Systems Multiple Bits s(t ) +1 t -1 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 72 / 174 Lecture Notes 2 Linear Systems Multiple Bits y (t ) t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 73 / 174 Lecture Notes 2 Linear Systems Multiple Bits y (t ) t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 74 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Noise Communications systems have some amount of noise. Electrical systems have random thermal noise due to motion of electrons because the system is not at absolute zero temperature. This noise is usually modeled as having power at all frequencies but in actuality at very high frequencies the power decreases (in the optical range of frequencies). The model widely used for thermal noise is that of zero mean white Gaussian noise. Since only the frequency band of the transmitted signal is of interest the noise outside this band is not important. For all systems considered here we model the noise as having equal power at all frequencies (white noise). In addition the noise will have zero mean or average. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 75 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Probability Because we can not know the value of the noise ahead of time we can only characterize it by its statistical properties such as average values. Using probability we can determine likelihoods of events. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 76 / 174 Lecture Notes 2 Random Variables...
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This note was uploaded on 02/12/2014 for the course EECS 455 taught by Professor Stark during the Fall '08 term at University of Michigan.

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