EC-23.5 Estimating the Channel, and Equalising It
You can explain how a known training sequence lets a receiver estimate the channel, derive the zero-forcing and minimum mean square error equalisers and say what each does to the noise, and describe why the frequency domain makes the problem easy.
Before:S5-3. Digital Signal ProcessingEC-11. Data Communications and Computer Networks
A channel smears each symbol into the ones after it, and the receiver has to undo that without knowing in advance what the channel was. The two halves of the job are estimation, which uses symbols both ends already know, and equalisation, which inverts what was estimated. The zero-forcing equaliser inverts exactly and amplifies noise where the channel was weak, which is the trade the whole topic is about.
Work through these
Intersymbol interference: what a dispersive channel does to a symbol stream
Energy from one symbol arrives during the next, so the sampled value depends on several transmitted symbols at once. Everything in this topic exists to undo that mixing.
NPTEL: Principles of Signal Estimation for MIMO/OFDM Wireless Communication (IIT Kanpur) · CourseTraining sequences and pilots, and the cost of sending them
Sending symbols the receiver already knows lets it solve for the channel, at the price of the capacity those symbols occupy. How often they are sent depends on how quickly the channel changes.
NPTEL: Principles of Signal Estimation for MIMO/OFDM Wireless Communication (IIT Kanpur) · CourseLeast squares estimation of the channel response
With a known input and an observed output, the channel is the solution of an ordinary overdetermined system. It is the same least squares calculation that appears in every other estimation problem in engineering.
NPTEL: Principles of Signal Estimation for MIMO/OFDM Wireless Communication (IIT Kanpur) · CourseThe zero-forcing equaliser, and where it amplifies noise
Inverting the channel exactly removes the interference and multiplies the noise by the inverse as well, which is disastrous at frequencies the channel attenuated. This is the canonical example of an exact answer being the wrong one.
NPTEL: Principles of Signal Estimation for MIMO/OFDM Wireless Communication (IIT Kanpur) · CourseThe minimum mean square error equaliser, and the compromise it makes
Minimising the total error rather than the interference alone leaves some interference in exchange for far less noise. It needs an estimate of the noise level, which is one more thing the receiver has to work out.
Why the frequency domain makes this easy, and what a cyclic prefix is for
Convolution becomes multiplication in frequency, so an equaliser becomes one complex division per subcarrier. A short repeated guard interval is what makes the channel behave as a circular convolution so that this actually holds.
NPTEL: Principles of Modern CDMA, MIMO and OFDM Wireless Communications (IIT Kanpur) · Course
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