# About the variance of the coefficient of narrowband noise when using signal-space representation

Assume that n(t) is a white Gaussian noise process with zero-mean and power spectrum density $$N_0/2$$. By using the signal-space representation, it can be expressed as:

$$n(t) =\sum_{j=1}^N n_j \phi_j(t)+n_o(t)$$

where $$n_j = \int_{-\infty}^{\infty}n(t)\phi_j(t)\,\mathrm dt$$ , $$\phi_j(t)$$ is the basis function and $$n_o(t)$$ is a part which cannot be expressed in terms of this basis function.

• If the noise process $$n(t)$$ is a narrowband noise of bandwidth $$B$$ centered on frequency $$f$$, can it still be expressed in this way?

My understanding is that only after the matched filter, the part $$n_o(t)$$ is removed. So the narrowband noise also can be expressed in this way.

• Is my understanding correct?

In addition, the coefficient $$n_j$$ must be a random variable, how to derive the variance of the $$n_j$$ in the case of narrowband noise ?

• the formula you've posted makes no sense! The sum includes $n_2$, which can by, the integral formula below, be expressed as coefficient to the basis function $\phi_2$. There's something wrong in the formula. – Marcus Müller Feb 24 at 9:11
• Sorry for my confused text expression. What I mean by that formula is the noise process cannot be completely expanded in terms of the basis functions. The first part of the noise process that can be expanded in terms of basis functions. The other part which cannot expressed in terms of this basis function is denoted by n2(t). – Berman Song Feb 24 at 9:49
• So, can we just rename $n_2$ to $n_o$, with $o$ like "orthogonal"? You can't just use $n_j$ and $n_2$ in a formula and say $n_j, j=2$ is not the same as $n_2$... – Marcus Müller Feb 24 at 10:57
• by the way, your first question doesn't make too much sense: you literally ask whether anything can be represented as a sum of something plus something else – yes, that is possible for everything. You will probably want to define how your $n_j$ and $\phi_j$ relate to your matched filter. – Marcus Müller Feb 24 at 18:08

Since the basis functions are narrowband, the formula still works and the variance of each $$n_j$$ remains $$N_0/2$$..
In other words, assume the bandwidth of the basis functions is $$B$$. You can then think of them as filters that ignore any frequency outside of $$B$$. So, they "don't care" if there is noise outside of $$B$$ or not.
The expression for $$n_0(t)$$ will change, though.
• I see your point. But, I still don't understand that if the noise $n(t)$ becomes a narrowband noise, and we calculate the covariance of $n_i$ and $n_j$ as $COV[n_in_j]=E[\int_{0}^{T}n(t)\phi_i(t)dt \int_{0}^{T}n(s)\phi_j(s)ds]=\int_{0}^{T} \int_{0}^{T} E[n(t)n(s)] \phi_i(t) \phi_j(s) dt ds$. Doesn't its autocorrelation function becomes $N_0Bsinc(\pi B\tau)cos(2 \pi f\tau)$ ? And by letting $\tau=0$, we can get the variance by solving the integral. Is that right? – Berman Song Mar 5 at 14:24