EC-23.5 Estimating the Channel, and Equalising It

What this is and why it exists

A channel smears each symbol into the ones that follow it, and the receiver has to undo that without knowing what the channel was.

The job splits in two. Estimation uses symbols both ends already know to work out what the channel did. Equalisation inverts it.

The interesting part is that inverting exactly is the wrong answer. The zero-forcing equaliser removes all the interference and amplifies the noise wherever the channel was weak. That trade is what this topic is really about.

The vocabulary

• Intersymbol interference — energy from one symbol arriving during another.
• Channel impulse response — what the channel does to a single isolated symbol.
• Training sequence — symbols the receiver already knows, sent to measure the channel.
• Pilot — a known symbol inserted periodically among the data.
• Least squares — fitting a model by minimising the total squared error.
• Zero-forcing — inverting the channel exactly, whatever it does to the noise.
• Noise enhancement — the amplification of noise caused by that inversion.
• Minimum mean square error — minimising total error rather than interference alone.
• Cyclic prefix — a repeated guard interval that makes convolution circular.

The mental model

A dispersive channel spreads each symbol in time. The value sampled at one instant therefore depends on several transmitted symbols, not one. That mixing is intersymbol interference, and everything else here exists to undo it.

To undo it you must first know it. Sending symbols the receiver already knows lets it compare what it received against what it expected, and solve for the channel. That costs capacity, and how often those symbols are sent depends on how quickly the channel changes. A stationary link needs them rarely; a moving one needs them constantly.

The solving itself is ordinary least squares. With a known input and an observed output, the channel response is the solution of an overdetermined system. The same calculation appears in every estimation problem in engineering, which is worth noticing.

Now the equaliser. Zero-forcing applies the exact inverse of the estimated channel. The interference vanishes completely. But wherever the channel attenuated a frequency, the inverse amplifies it, and the noise at that frequency is amplified with it. On a channel with a deep notch, the result is worse than doing nothing.

Minimum mean square error equalisation minimises the total error instead. It leaves some interference in exchange for far less noise amplification, and the balance depends on the noise level. That means the receiver has to estimate the noise as well, which is one more unknown.

Finally, why the frequency domain helps so much. Convolution in time becomes multiplication in frequency, so an equaliser becomes one complex division per subcarrier. That only holds exactly if the channel acts as a circular convolution. A short repeated guard interval at the start of each block is what arranges it.

What you should now be able to explain or do

• Explain intersymbol interference in terms of the channel impulse response.
• Say what a training sequence costs and what decides how often it is sent.
• Estimate a channel by least squares from known transmitted symbols.
• Derive the zero-forcing equaliser and show where it amplifies noise.
• Say what the minimum mean square error equaliser trades and what it needs.
• Explain why a cyclic prefix makes per-subcarrier equalisation valid.

Check yourself