1. 3F4 Optimal Transmit and Receive Filtering Dr. I. J. Wassell.

2. Transmission System The FT of the received pulse is, Transmit Filter, H T (  ) Channel H C (  ) Receive Filter, H R (  ) + N(  ), Noise Weighted impulse train To slicer y(t)

3. Transmission System Where, –H C (  ) is the channel frequency response, which is often fixed, but is beyond our control –H T (  ) and H R (  ), the transmit and receive filters can be designed to give best system performance How should we choose H T (  ) and to H R (  ) give optimal performance?

4. Optimal Filters Suppose a received pulse shape p R (t) which satisfies Nyquist’s pulse shaping criterion has been selected, eg, RC spectrum The FT of p R (t) is P R (  ), so the received pulse spectrum is, H(  )=k P R (  ), where k is an arbitrary positive gain factor. So, we have the constraint,

5. Optimal Filters For binary equiprobable symbols, Where, V o and V 1 are the received values of ‘0’ and ‘1’ at the slicer input (in the absence of noise)  v is the standard deviation of the noise at the slicer Since Q(.) is a monotonically decreasing function of its arguments, we should,

6. Optimal Filters x f(x) b 0 0 b Q(b) 0.5 1.0

7. Optimal Filters For binary PAM with transmitted levels A 1 and A 2 and zero ISI we have, Remember we must maximise, Now, A 1, A 2 and p R (0) are fixed, hence we must,

8. Optimal Filters Noise Power, –The PSD of the received noise at the slicer is, –Hence the noise power at the slicer is, HR()HR() n(t)v(t) N ()N ()Sv()Sv()

9. Optimal Filters We now wish to express the gain term, k, in terms of the energy of the transmitted pulse, h T (t) From Parsevals theorem, We know,

10. Optimal Filters So, Giving, Rearranging yields,

11. Optimal Filters We wish to minimise,

12. Optimal Filters Schwartz inequality states that, With equality when, Let,

13. Optimal Filters So we obtain, All the terms in the right hand integral are fixed, hence,

14. Optimal Filters Since is arbitrary, let  =1, so, Utilising, And substituting for H R (  ) gives, Receive filter Transmit filter

15. Optimal Filters Looking at the filters –Dependent on pulse shape P R (  ) selected –Combination of H T (  ) and H R (  ) act to cancel channel response H C (  ) –H T (  ) raises transmitted signal power where noise level is high (a kind of pre-emphasis) –H R (  ) lowers receive gain where noise is high, thereby ‘de-emphasising’ the noise. Note that the signal power has already been raised by H T (  ) to compensate.

16. Optimal Filters The usual case is ‘white’ noise where, In this situation, and

17. Optimal Filters Clearly both |H T (  )| and |H R (  )| are proportional to, ie, they have the same ‘shape’ in terms of the magnitude response In the expression for |H T (  )|, k is just a scale factor which changes the max amplitude of the transmitted (and hence received) pulses. This will increase the transmit power and consequently improve the BER

18. Optimal Filters If |H C (  )|=1, ie an ideal channel, then in Additive White Gaussian Noise (AWGN), That is, the filters will have an identical RC 0.5 (Root Raised Cosine) response (if P R (  ) is RC) Any suitable phase responses which satisfy, are appropriate

19. Optimal Filters In practice, the phase responses are chosen to ensure that the overall system response is linear, ie we have constant group delay with frequency (no phase distortion is introduced) Filters designed using this method will be non- causal, i.e., non-zero values before time equals zero. However they can be approximately realised in a practical system by introducing sufficient delays into the TX and RX filters as well as the slicer

20. Causal Response Note that this is equivalent to the alternative design constraint, Which allows for an arbitrary slicer delay t d, i.e., a delay in the time domain is a phase shift in the frequency domain.

21. Causal Response Non-causal response T = 1 s Causal response T = 1s Delay, t d = 10s

22. Design Example Design suitable transmit and receive filters for a binary transmission system with a bit rate of 3kb/s and with a Raised Cosine (RC) received pulse shape and a roll-off factor equal to 1500 Hz. Assume the noise has a uniform power spectral density (psd) and the channel frequency response is flat from -3kHz to 3kHz.

23. The channel frequency response is, Design Example H C (f) H C (  ) f (Hz)  rad/s) 0 3000 2  3000 -3000 -2  3000

24. Design Example The general RC function is as follows, P R (f) f (Hz) T 0

25. Design Example For the example system, we see that  is equal to half the bit rate so,  =1/2T=1500 Hz Consequently, 0 P R (f) f (Hz) T 15003000  (rad/s)2  15002  00

26. Design Example So in this case (also known as x = 1) where, We have, Where both f and  are in Hz Alternatively, Where both  and  are in rad/s

27. Design Example The optimum receive filter is given by, Now N o and H C (  ) are constant so,

28. Design Example So, Where a is an arbitrary constant. Now, Consequently,

29. Design Example Similarly we can show that, So that,

30. Summary In this section we have seen –How to design transmit and receive filters to achieve optimum BER performance –A design example