EECS20N: Signals and Systems

# The Fourier series, using complex exponentials

The Fourier series for a continuous-time, periodic signal has been given as

x(t) = A0 + ∑(k=1 to ∞) Ak cos (kω0t + φ k )

where x(t) is a periodic function with period p = 2π /ω 0.

For reasons that we can now understand, the Fourier series is usually written in terms of complex exponentials rather than cosines. Since complex exponentials are eigenfunctions of LTI systems, this form of the Fourier series decomposes a signal into components that when processed by the system are only scaled.

Euler's formula states that

exp(jθ ) = cos(θ ) + j sin(θ ).

The complex conjugate is

exp(− jθ ) = cos(θ ) − j sin(θ )

Summing these,

exp(jθ ) + exp(− ) = 2cos(θ )

or

cos(θ ) = 0.5(exp(jθ ) + exp(− jθ )).

Each term of the Fourier series expansion has the form

Ak cos(kω 0t + φphi;phi; k)

which we can write

Ak cos(kω 0t + φphi;phi; k) = 0.5 Ak (exp(j(kω 0t + φphi;phi; k)) + exp(− j( kω 0t + φphi;phi; k))).

so the Fourier series can be written

x(t) = A0 + ∑(k=1 to ∞) 0.5Ak (exp ( j(kω0t + φ k )) + exp (− j(kω0t + φ k ))).

Observe that exp( j(0t + φphi;phi; k)) = exp( jkω 0t)exp(k) and let

X0 = A0

Xk = 0.5 Ak exp(k), k = 1, 2, …

X-k = Xk* = 0.5 Ak exp(− k), k = 1, 2, …

then the Fourier series becomes

x(t) = ∑(k= − ∞ to ∞) Xk exp ( jkω0t) .

This is the form in which one usually sees the Fourier series.

The discrete-time Fourier series can be similarly written

x(n) = ∑(k=0 to p−1) Xk exp ( jkω0n) .

There are some important differences in this case however. First and foremost, the sum is finite, making it manageable by computer. Second, it's exact for any periodic waveform.  There are no mathematically tricky cases, and no approximation needed.