First Differentatior and Integrator

Question

For discrete time series $Y$, first differentiation ($D_i=Y_i-Y_{i+1}$) and integrator ($S_i=Y_i+Y_{i+1}$) can be defined as two highpass and lowpass LTI digital filters. Where, transfer function for the first difference is: $$ H(z)=1-z^{-1} $$ and for the first integrator is: $$ H(z)=1+z^{-1} $$ These two transfer function corresponds to $\text{Conv}(Y,[1,-1])$ and $\text{Conv}(Y,[1,1])$ for differentiator and integrator, respectively.

I was wondering what the power transfer function of these two filters should be.

I have to add, in terms of implementation in Matlab, I can get $S_i$ and $D_i$ of a white noise by:

Y = randn(1,100); D = diff(Y); S = Y(1:end-1)+Y(2:end);

By filter in Matlab:

S_Filt = filter([1 1],1,Y); D_Filt = filter([1 -1],1,Y);

Or by convolving conv the corresponding windows: S_Conv = conv(Y,[1 1]); D_Conv = conv(Y,[1 -1]);

Note, all D* and S* variables should be identical (regardless of head and tail of time series, of course).

Thanks in advance.

Answer 1

I think I found an answer to this question. So, I will post it here.

For power transfer function of $1-z^{-1}$ we know: \begin{eqnarray*} 1-z^{-1}&=&1-e^{-jw}\\ &=& 1-\cos(\omega)+j\sin(w) \end{eqnarray*} Then by using complex conjugate properties we have: \begin{eqnarray*} |1-z^{-1}|^2&=&1-2\cos(\omega)+\cos(\omega)^2+\sin(\omega)^2\\ &=&2(1-\cos(\omega)). \end{eqnarray*} And simillarly, for $1+z^{-1}$, we have: \begin{eqnarray*} |1+z^{-1}|^2&=& 1+2\cos(\omega)+\cos(\omega)^2+\sin(\omega)^2\\ &=& 2(1+\cos(\omega)). \end{eqnarray*}

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Note that the integrator in this question is a special case (see the expression for $S_i$) and is different than the conventional integrator $y[n]=y[n-1]+x[n]$.