3F4 Optimal Transmit and Receive Filtering

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3F4 Optimal Transmit and Receive Filtering

• Dr. I. J. Wassell.

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Transmission System
Transmit Filter, HT(w)
Channel HC(w)
Receive Filter, HR(w)
y(t)

Weighted impulse train
To slicer
N(w), Noise

• The FT of the received pulse is,

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Transmission System

• Where,
• HC(w) is the channel frequency response, which is
  often fixed, but is beyond our control
• HT(w) and HR(w), the transmit and receive filters
  can be designed to give best system performance
• How should we choose HT(w) and to HR(w) give
  optimal performance?

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Optimal Filters

• Suppose a received pulse shape pR(t) which
  satisfies Nyquists pulse shaping criterion has
  been selected, eg, RC spectrum
• The FT of pR(t) is PR(w), so the received pulse
  spectrum is, H(w)k PR(w), where k is an
  arbitrary positive gain factor. So, we have the
  constraint,

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Optimal Filters

• For binary equiprobable symbols,

• Where,
• Vo and V1 are the received values of 0 and 1
  at the slicer input (in the absence of noise)
• sv is the standard deviation of the noise at the
  slicer

• Since Q(.) is a monotonically decreasing
  function
• of its arguments, we should,

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Optimal Filters
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Optimal Filters

• For binary PAM with transmitted levels A1 and A2
  and zero ISI we have,

• Remember we must maximise,

Now, A1, A2 and pR(0) are fixed, hence we must,
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Optimal Filters

• Noise Power,
• The PSD of the received noise at the slicer is,

• Hence the noise power at the slicer is,

n(t)
v(t)
HR(w)
N (w)
Sv(w)
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Optimal Filters

• We now wish to express the gain term, k, in terms
  of the energy of the transmitted pulse, hT(t)

• From Parsevals theorem,

• We know,

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Optimal Filters

• So,

• Giving,

• Rearranging yields,

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Optimal Filters

• We wish to minimise,

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Optimal Filters

• Schwartz inequality states that,

With equality when,
Let,
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Optimal Filters

• So we obtain,

All the terms in the right hand integral are
fixed, hence,
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Optimal Filters

• Since l is arbitrary, let l1, so,

Receive filter

• Utilising,

And substituting for HR(w) gives,
Transmit filter
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Optimal Filters

• Looking at the filters
• Dependent on pulse shape PR(w) selected
• Combination of HT(w) and HR(w) act to cancel
  channel response HC(w)
• HT(w) raises transmitted signal power where noise
  level is high (a kind of pre-emphasis)
• HR(w) lowers receive gain where noise is high,
  thereby de-emphasising the noise. Note that the
  signal power has already been raised by HT(w) to
  compensate.

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Optimal Filters

• The usual case is white noise where,

• In this situation,

and
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Optimal Filters

• Clearly both HT(w) and HR(w) are proportional
  to,

ie, they have the same shape in terms of the
magnitude response

• In the expression for HT(w), 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

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Optimal Filters

• If HC(w)1, ie an ideal channel, then in
  Additive White Gaussian Noise (AWGN),

• That is, the filters will have an identical RC0.5
  (Root Raised Cosine) response (if PR(w) is RC)

• Any suitable phase responses which satisfy,

are appropriate
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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

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Causal Response

• Note that this is equivalent to the alternative
  design constraint,

Which allows for an arbitrary slicer delay td ,
i.e., a delay in the time domain is a phase shift
in the frequency domain.
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Causal Response
Non-causal response T 1 s
Causal response T 1s Delay, td 10s
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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.

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Design Example

• The channel frequency response is,

HC(f) HC(w)
0
f (Hz) w (rad/s)
3000 2p3000
-3000 -2p3000
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Design Example

• The general RC function is as follows,

PR(f)
T
0
f (Hz)
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Design Example

• For the example system, we see that b is equal to
  half the bit rate so, b1/2T1500 Hz
• Consequently,

PR(f)
T
0
f (Hz)
1500
3000
w(rad/s)
2p1500
2p3000
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Design Example

• So in this case (also known as x 1) where,

• We have,

Where both f and b are in Hz

• Alternatively,

Where both w and b are in rad/s
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Design Example

• The optimum receive filter is given by,

• Now No and HC(w) are constant so,

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Design Example

• So,

Where a is an arbitrary constant.

• Now,

• Consequently,

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Design Example

• Similarly we can show that,

• So that,

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Summary

• In this section we have seen
• How to design transmit and receive filters to
  achieve optimum BER performance
• A design example