# Convolution between two unequal length vectors

Considering an example, where the model is \begin{align} x[n] = \mathbf{h}^\mathsf{T}\mathbf{u}[n] + w[n]. \label{Eq1} \end{align} $\mathbf{h} = [h_0,h_1,\ldots,h_{M-1}]^\mathsf{T}$ of length $M$ which represents the coefficients of the LTI SISO channel and $\mathbf{u}[n] = [u[n], u[n-1], u[n-2],\ldots,u[n-M+1]$ is the input to the channel. Let $\mathbf{v} = [v_0,v_1,\ldots,v_{M-1}]^\mathsf{T}$ of length $M$ be the coefficients of the equalizer.

If the length of the equalizer is different, say the channel length is $M$ and equalizer length is $Q=2M-1$ then how would the convolution happen since there will be a mismatch in the number of elements of $\mathbf{h}$ and $\mathbf{v}$

I can do conv(h,w) in Matlab where h is the coefficients of the channel and w denotes the coefficients of the equalizer. Assuming known values of h = [1,0.2,0.6] and the estimated coefficients of the equalizer as `w = [0.8,0.1,0.2,0.3,0.5]'. How would the convolution work? Can somebody please show first few steps. The first few sample will be zero based on my calculation. But i am not sure.

• Your question is not clear. First what is the difference between $w[n]$ and your model and your equalizer w ? I suppose you meant $v$ as equalizer ? Second, what is the input and output of your equalizer ? Commented Nov 19, 2017 at 19:43