Is there any simple way or trick to express given signal using waveforms of sum of simple sinusoidal signals?
Take the shortest repeating interval of the waveform - that seems to be one-quarter of what you have drawn - that is the time period of the fundamental frequency and, to find the amplitude of that fundamental, multiply that section of signal by a sine wave and a cosine wave of the same time period.

Then integrate (over the time period) the two multiplied waveforms to get two numbers. Divide those two numbers by the time period and you get the a and b coefficients that pertain to the fundamental signal. Well actually you get the RMS values so multiply them by 1.4142 to get the true a and b coefficients.

Repeat for the 2nd harmonic and keep going up in harmonics until you are satisfied there is no appreciable signal energy left to worth considering.

You can do it in excel if you have sample values for the repeating signal. If all you have is a picture then you are out of luck.