I'm trying to express the following set of signals on orthonormal basis functions. These signals are to be transmitted using a 4-ary modulation scheme.How do i write ψ1(t)and ψ2(t) for the same. Also how to find orthonormal function set for given set of signals.

These are the given four signals

  • $\begingroup$ You need to apply the Gram-Schmidt algorithm, which is probably described in the same textbook the problem came from. If you have a specific problem applying the algorithm, please add it to your question. $\endgroup$ – MBaz Mar 5 '15 at 12:51
  • $\begingroup$ Hey! Thanks. This is a problem I found in old notes of my senior. Can you pls name the book you are referring to? B. Sklar? $\endgroup$ – user12083 Mar 5 '15 at 12:58
  • $\begingroup$ I'm sure it's in Sklar, but I don't have it at hand right now. It's also in Wikipedia. $\endgroup$ – MBaz Mar 5 '15 at 14:37
  • $\begingroup$ Here is a note (hopefully accessible world-wide) that I wrote a long time ago describing the Gram-Schmidt process. $\endgroup$ – Dilip Sarwate Mar 5 '15 at 14:57

Let's call your $4$ signals $f_0$, $f_1$, $f_2$ & $f_3$. Just by eyeballing you could have a basis of two function where

b0 = f0, for 0 < t < Ts/2, 0 otherwise and
b1 = f0, for Ts/2 < t < Ts, 0 otherwise and

So you basically have a one first half ramp up in the first half and one second half ramo down. The signals can be expressed simply as

f0 =  b0 + b1
f1 = -b0 - b1
f2 =  b0 - b1
f3 = -b0 + b1

So it's clearly a base. It's also orthogonal since $b_0 b_1$ is zero for all $t$. It's probably not normal but could be easily made so by proper scaling.

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Even more simpler than @Hilmar's answer, the signals $f_0$ and $f_2$ are orthogonal just by eyeballing the pictures and recalling basic ideas about integration giving the area under the curve. Thus, the signals are $f_0, -f_0, f_2, -f_2$ with respect to the orthogonal basis $\{f_0, f_2\}$. The basis could be made a orthonormal basis (unit-energy basis signals) if absolutely needed.

Interestingly, the above description shows that the four signals are a bi-orthogonal signal set -- a $4$-QAM signal set rotated by $\pi/4$ in signal space -- for which the symbol error probabilities and bit error probabilities have been studied and documented throroughly. The bi-orthogonality is not as easily noticed with the basis used in Hilmar's answer.

As a minor note, if we use the Gram-Schmidt orthogonalization procedure (as suggested by @MattL in a comment) starting with $f_0$ and continuing with $f_1, f_2, f_3$, we end up with $\{f_0, f_2\}$ as the orthogonal basis for this set of $4$ signals.

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