Equalizers in Wireless Communication
Typical wireless communication introduce multipath fading that leads to ISI. Estimating the channel is necessary to compensate for these effects. By sending a known prefix alongside the data, the channel response can be determined using Fourier-transform-based methods:
H(f) = Pr(f) / Pt(f)
h(t) = IFFT(H(f))
Here, Pr(f) and Pt(f) are the Fourier transforms of the received and transmitted prefixes. Although this approach is straightforward, it is sensitive to noise.
Equalization and ISI Mitigation
Pulses transmitted through underwater channels often get distorted, producing inter-symbol interference. Equalizers are used to counteract this effect. Adaptive equalizers, such as Recursive Least Squares (RLS) or Least Mean Squares (LMS), adjust their parameters based on the channel's characteristics.
Using TRM can simplify the channel by reducing eigenvalue spread, which improves equalizer convergence. Combining TRM with adaptive equalization minimizes bit errors and improves signal accuracy.
Why Equalizers Are Needed
Wireless channels distort signals due to:
- Multipath propagation → Inter-Symbol Interference (ISI)
- Frequency-selective fading → some frequencies attenuated more
- Noise → Additive White Gaussian Noise (AWGN)
Received signal model:
r(t) = s(t) * h(t) + n(t)- s(t): transmitted signal
- h(t): channel impulse response
- n(t): noise
Goal of the equalizer: Recover s(t) from r(t) by compensating for the channel h(t).
Mathematical Model of a Simple Wireless Equalizer
r[n] = Σ (h[k] * s[n-k]) + n[n], k = 0..L-1The equalizer applies a filter w[m] to estimate s[n]:
ŝ[n] = Σ (w[m] * r[n-m]), m = 0..M-1Goal: Minimize Mean Square Error (MSE):
min_w E[ |s[n] - ŝ[n]|² ]Types of Equalizers
Linear Equalizer
- Simple FIR filter w[m]
- Zero-Forcing (ZF) equalizer: W_ZF = H⁻¹
- Disadvantage: amplifies noise in weak channel frequencies
Minimum Mean Square Error (MMSE) Equalizer
- Minimizes MSE considering noise
W_MMSE = (Hᴴ H + σ_n² I)⁻¹ HᴴDecision Feedback Equalizer (DFE)
- Uses previous detected symbols to cancel ISI
- Combines feedforward and feedback filters
Frequency-Domain View
If the channel is frequency-selective:
R(f) = H(f) S(f) + N(f)Frequency-domain equalization:
Ŝ(f) = W(f) * R(f)This is similar to audio equalizers: shape the frequency response to recover the original signal.
Simple Example: 2-Tap Channel
Channel: h[0] = 1, h[1] = 0.5
r[n] = s[n] + 0.5 s[n-1] + n[n]Linear equalizer coefficients w[0], w[1] chosen such that:
ŝ[n] = w[0] r[n] + w[1] r[n-1] ≈ s[n]Solution via MSE minimization approximately recovers s[n].
Summary
- Equalizers undo channel distortion.
- Crucial for multipath channels and frequency-selective fading.
- Can be time-domain (FIR/IIR) or frequency-domain (FFT-based).
- Trade-off between ISI reduction and noise enhancement (ZF vs MMSE).
- Often combined with adaptive algorithms (LMS, RLS) in time-varying channels.
- Wireless channels distort signals → equalizers restore them.
- Discrete-time model: ŝ[n] = Σ w[m] r[n-m]
- Linear equalizer: direct FIR filter
- MMSE equalizer: balances noise and ISI
- Frequency-domain equalizer: multiplies by 1/H(f)
- DFE: cancels ISI using past decisions
In general wireless communication systems are modeled as linear time-invariant (LTI) systems. The received signal is considered the convolution of a transmitted signal and channel input response (CIR) in the time domain. In the frequency domain, we observe a slight frequency shift. To retrieve the original signal at the receiver side, we need to go through the 'deconvolution' process. There the no standard process named 'deconvolution' in the case of wireless communication. The equalization process does the same job.