Euler’s identity is very useful not only in obtaining the rectangular and polar forms of complex numbers, but in many other respects as we will explore in this problem.

(a) Carefully plot x[n] = e^jπn for integers − ∞ < n < ∞. Is this a real or a complex signal?

(b) Suppose you want to find the trigonometric identity corresponding to sin(α) sin(β). Use Euler’s identity to express the sines in terms of exponentials, multiply the resulting exponentials and use Euler’s identity to regroup the expression in terms of sinusoids.

(c) As we will see later on, two periodic signals x(t) and y(t) of period T₀ are said to be orthogonal if the integral over a period T₀

For instance, let x(t) = cos(πt) and y(t) = sin(πt). Check first that these functions repeat every T₀ = 2, i.e., show that x(t + 2) = x(t) and that y(t + 2) = y(t). Thus T₀ = 2 can be seen as their fundamental period. Use Euler’s identity to express the integrand in terms of exponentials and verify the integral is zero or that x(t) and y(t) are orthogonal. You may use symbolic MATLAB to verify the integral is zero.

Transcribed Image Text:

x(t)y(t)dt = 0 JT.