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Function of a Decision Feedback Equalizer (DFE) Equalizer (with MATLAB)

 

Decision Feedback Equalizer is the abbreviation for this. We know that in a typical wireless communication scenario, different multipath cause the signal to arrive at the receiver at different times after transmitting it from the transmitter. Our signal may show a slight spatial frequency shift as a result. By using several taps to receive signals with varying time delays or signals with a slight frequency shift, equalizers solve this problem by changing their tap weights.

Decision feedback equalizers continuously update their tap coefficient vectors by reducing the error between the desired signal and adaptive filter output.

Inter-symbol interference (ISI) happens in a typical wireless communication system when the modulation bandwidth exceeds the radio channel's coherence bandwidth. Equalization reduces the ISI produced by multipath within a time-dispersive channel.

** Coherence bandwidth is the bandwidth (range of frequencies) over which the channel is constant is called coherence bandwidth

Both feed-forward taps and feed-backward taps are used in a DFE equalizer. 

'Input' - Input signal, specified as a column vector
'Desired' - Training symbols (column vector). The vector length of the desired must be less than or equal to the length of the input.
'Train' - Train equalizer flag. It starts training when the value changes from 0 to 1.
'Error' - Error signal (column vector)
'Weight' - Tap weights

The error between the feedforward and feedback signal is used to adjust the tap weights. To recover the original signal, it also converges the tap weights. DFE uses an adaptive algorithm at the receiver side to track the changing channel and modifies the weight of its filter to recover the known pilot signal most of the time. The data bits are then appropriately corrected. Pilot signals are provided at short intervals and are known to the receiver to track the communication channel or medium's frequently changing characteristics.

 


QPSK + Multipath + Decision Feedback Equalizer (DFE)

20 BER Before: 0 | BER After: 0

Simulator Workflow & Mathematics

1 The Signal Path

  1. QPSK Modulation Random bits are grouped into pairs and mapped to complex symbols: s = exp(j * (π/4 + kπ/2)).
  2. Multipath Channel The signal is convolved with a multi-tap impulse response, simulating reflections that cause Inter-Symbol Interference (ISI).
  3. DFE Equalization A Feedforward Filter (FFF) suppresses precursor ISI, while a Feedback Filter (FBF) subtracts post-cursor ISI using previous symbol decisions.

2 The DFE Equations

// 1. Equalizer Output

y[n] = Σ(w_f[k] * x[n-k]) - Σ(w_b[k] * d[n-k])

(Feedforward Output - Feedback Correction)

// 2. Error Calculation

e[n] = d_known[n] - y[n]

// 3. LMS Weight Update

w_f[k] = w_f[k] + μ * e[n] * x*[n-k]

w_b[k] = w_b[k] - μ * e[n] * d*[n-k-1]

Where μ is the step size, x* is the conjugate of the received signal, and d are previous decisions.
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Also read about

[2] Adaptive Equalizer to mitigate Channel Distortion (in MATLAB)

[3] Roll of an Equalizer in Channel Estimation


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