Assume we have downconverted a received signal via a mixing operation and now we wish to apply a LPF. One way to implement a I.PF is simply to compute an average. In continuous time we would just integrate the signal since dividing by the integration time T to compute the average would not affect the performance since the noise would be scaled by the same amount as the signal component . In discrete time we would implement a sum instead of an integral. For this part we will assume we have a discrete time signal but we will compute an average instead of just a sum.
So let us assume the input to the LPF is a signal of the form
s(k) = fË + double frequency terms + n(k), k = 0,1,…
where we have assumed scaling so that n(k) is a standard normal random variable for each k and n, is independent of rij for i ^ j, i,j = 0,1,…. We may ignore the double frequency terms and assume they are suppressed, either completely or at least sufficiently, by the LPF. The output of the LPF

is
k=n-N+l
where we take y(n) = 0 for n <0. Even though we are computing an average we will refer to this type of filter as an integrate and dump or I&D filter. Now for implementation purposes we can also construct a LPF using an HR filter. Let