# Beginner level : Help with terminologies : smooting length, length of channel, channel equalization delay, blind system identification, equalization

Say the channel model is an univariate FIR filter with true coefficients $h=[1,a_1, a_2 ]$. I am learning algorithms for system identification and channel equalization. For this, I am implementing programs provided in Matlab. One such Algorithm is Constant Modulus Algorithm for blind channel equalization.

Code is the link for the code : lines 80-82 present the variables $L,ChL,EqD$ which denotes the smoothing length, length of the channel & channel equalization delay respectively. I have confusion regarding the terms. Shall appreciate explanation with an example pertaining to my case.

1. Smoothing length : In text books, $L$ stands for the channel order, which for my model, Moving Average, MA(3), the order then = 3 ( I guess including the 1). FOr Autoregressive model AR(p) $p$ is the model order. I am confused if smoothing length has got anything to do with Kalman Smoother ? What is the difference, if any, between smoothing length and channel order? What will $L$ be for the FIR model that I have?

2. Length of the Channel, ChL = Is this related to number of data samples ?

3. Equalization delay, EqD = In general, I was under the impression that the model order is determined by the number of delays. So for MA(q) : $q$ is the number of delays, similarly for AR(p) : $p$ is the number of delays. Then what is Equalization delay?

1. When considering blind algorithm, we assume no knowledge of the process noise that drives the model. But, When we assume that the model order is known, then does it still become blind or non-blind estimation / equalization technique?

The term smoothing length (also called observation interval) is simply the length of the equalizer, i.e. its number of filter taps. In the code the filter length is L+1, so L is the filter order. Similarly, the length of the channel refers to the length of the FIR filter simulating the dispersive channel. Again, the channel filter length is actually ChL+1. What is called equalization delay in the code is the total delay introduced by the channel and by the equalizer. It is needed for aligning transmitted and received (and equalized) symbols.
• Thank you for your answer. Just to confirm if my understanding is right, can you please check this? The channel is modeled as a FIR filter (a) Let the FIR filter be $u(t)=ϵ(t)+h 1 ϵ(t−1)+h 2 ϵ(t−2)$ then filter order = number of delays = 2.; length of the filter = length of the channel = smoothing length L+1 = 3 ? What is the reason for considering smoothing length & length of FIR = L+1? It is yet unclear (b) Also, what is the equalization delay? –
• @SKM: No, the channel length is not equal to the smoothing length. As I said in my answer, the smoothing length is the length of the equalizer, not of the channel. There are two FIR filters in the simulation, one simulates the channel, and the other one is the actual equalizer implemented in the receiver. The equalization delay is just the delay introduced by the channel and by the equalizer. So if the transmitter sends a symbol at time instance $n$, then the corresponding symbol is detected by the receiver at $n+EqD$. May 23 '15 at 9:27