2.5.3 Complex Signal Space and Orthogonality - Modern Digital and Analogue Communication Systems - Lathi and Ding
Representing a function g(t) by another function x(t) in signal space over the interval [t1,t2].The error is:
\begin{array}{c} e( t) =g( t) -cx( t) \end{array}
To minimise the error, we need to select c to be the optimal value. Therefore need to minimise:
\begin{array}{l} E_{e} \ =\int _{t1}^{t2}| g( t) -cx( t)| ^{2} dt \end{array}
Where g(t) and x(t) are complex.
Using the following "identities"
\begin{array}{l} E_{x} =\int _{t1}^{t2}| x( t)| ^{2} dt\\ \end{array}
Where Ex is just the signal energy, analogues to the dot product of a vector. And:
\begin{array}{l} | u+v| ^{2} =( u+v)\left( u^{*} +v^{*}\right) =| u| ^{2} +| v| ^{2} +u^{*} v+uv^{*} \end{array}
Now he skips a bunch of steps and comes up with the following expansion:
\begin{array}{l} E_{e} \ =\int _{t1}^{t2}| g( t)| ^{2} \ dt-\left| \frac{1}{\sqrt{E_{x}}}\int _{t1}^{t2} g( t) x^{*}( t) \ dt\right| ^{2} +\left| c\sqrt{Ex} -\frac{1}{\sqrt{E_{x}}}\int _{t1}^{t2} g( t) x^{*}( t) \ dt\right| ^{2} \end{array}
How?
I can get it to the following:
\begin{array}{l} E_{e} \ =\int _{t1}^{t2}| g( t)| ^{2} \ dt-c\int _{t1}^{t2} g( t) x^{*}( t) \ dt-c\int _{t1}^{t2} g^{*}( t) x( t) \ dt+c^{2}\int _{t1}^{t2}| x( t)| ^{2} \ dt\\ \end{array}
and I know Ex so I can substitute this in:
\begin{array}{l} E_{e} \ =\int _{t1}^{t2}| g( t)| ^{2} \ dt-c\int _{t1}^{t2} g( t) x^{*}( t) \ dt-c\int _{t1}^{t2} g^{*}( t) x( t) \ dt+c^{2} E_{x}\\ \end{array}
Thanks