The complex exponential (or two-sided) Fourier series General information and Example 2.5

Two-Sided or Complex Exponential Form of the Fourier Series Advantages: Treats dc term the same as all other terms Complex form is used to derive Fourier transform for nonperiodic waveforms (discussed later) Complex form is also the basis for the discrete Fourier transform (continued)

Two-Sided or Complex Exponential Form (continued) Disadvantages: Form is not intuitive due to Use of complex exponentials Complex cn coefficients Resultant “negative frequency” components (continued)

Plotting cn in the Complex Plane Since Euler’s identity produces a real cosine term and an imaginary sine term, the cn coefficients of the two-sided Fourier series are also complex numbers. We can plot such a number in the complex plane (real portion represented by the x axis, imaginary portion represented by the y axis).

Plotting cn in the Complex Plane fn cn Imaginary axis Real axis Im{cn} Re{cn} We can express cn in terms of a real and imaginary component, or in terms of a magnitude |cn| and a phase fn. From our earlier work, we know how to relate cn and fn to real-world quantities, the phase and magnitude of the one-sided form: for n = 0, cn = Xn for n > 0, cn =0.5 Xn and phase = fn for n < 0, cn =0.5 X-n and phase = - f-n

Example 2.5 Use the complex exponential form of the Fourier series to represent the signal x(t) shown in Examples 2.3 and 2.4 (reproduced below). Draw the two-sided magnitude and phase spectra of the signal. volts x(t) 1 2 3 · · · · · · -5 -10 5 10 seconds

Solution to Example 2.5 In Example 2.4 we determined the Xn and fn coefficients of the one-sided form of the Fourier series for x(t). As shown earlier, for n = 0, cn = Xn for n > 0, cn =0.5 Xn and phase = fn for n < 0, cn =0.5 X-n and phase = - f-n The two-sided magnitude and phase spectra are thus plotted below: 0.6 0.5 0.4 0.3 0.2 0.1 Frequency in Hz Magnitude in volts 180 Phase in degrees 135 90 45 1 2 3 -3 -2 -1 1 3 2 -3 -2 -1 -45 -90 -135 -180 Frequency in Hz

Interpreting the “Negative Frequency” Components Produced by the Two-Sided Fourier Series The magnitude and phase spectra produced by the one-sided form of the Fourier series have physical meaning. The magnitude and phase spectra produced by the two-sided form of the Fourier series do not have physical meaning per se. “Negative frequency” does not exist in the real world — it is just a mathematical concept needed to correlate the one-sided and two-sided forms of the Fourier series. The “negative frequency” components of the two-sided Fourier series (corresponding to the summation from n = - to n = -1) physically represent additional contributions at the corresponding positive frequency. (continued)

Interpreting the “Negative Frequency” Components ... (continued) When determining real-world magnitude or power, you must therefore consider (i.e., add) both the positive and corresponding “negative” frequency components. You can ignore the phase of the two-sided “negative frequency” components.

Channel with 1Hz bandwidth passes first five harmonics sec -2 -1 1 2 3 4 5 6 7 8 -0.5 0.5 1.5 2.5 volts sec -2 -1 1 2 3 4 5 6 7 8 -0.5 0.5 1.5 2.5 volts Channel with 1Hz bandwidth 0.6 0.5 0.4 0.3 0.2 0.1 Frequency in Hz Magnitude in volts

Channel with 2Hz bandwidth passes first ten harmonics sec -2 -1 1 2 3 4 5 6 7 8 -0.5 0.5 1.5 2.5 volts sec -2 -1 1 2 3 4 5 6 7 8 -0.5 0.5 1.5 2.5 volts Channel with 2Hz bandwidth 0.6 0.5 0.4 0.3 0.2 0.1 Frequency in Hz Magnitude in volts

Channel with 3Hz bandwidth passes first fifteen harmonics sec -2 -1 1 2 3 4 5 6 7 8 -0.5 0.5 1.5 2.5 volts sec -2 -1 1 2 3 4 5 6 7 8 -0.5 0.5 1.5 2.5 volts Channel with 3Hz bandwidth 0.6 0.5 0.4 0.3 0.2 0.1 Frequency in Hz Magnitude in volts