Find the trigonometric Fourier series for the sawtooth wave shown in Fig. 17-39 and plot the line spectrum. Compare with Example 17.1.
The Correct Answer and Explanation is :
The Fourier series representation of a sawtooth wave is given by:
[
f(t) = \sum_{n=1}^{\infty} a_n \sin(n\omega_0 t) + b_n \cos(n\omega_0 t)
]
For a sawtooth wave with a period ( T ) and amplitude range from ( -A ) to ( A ), the Fourier series takes the form:
[
f(t) = \frac{A}{2} – \frac{2A}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin(n\omega_0 t)
]
where the fundamental angular frequency is ( \omega_0 = \frac{2\pi}{T} ).
Step 1: Compute Fourier Coefficients
The general form of a Fourier series is:
[
f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right)
]
For a sawtooth wave (odd function), ( a_n = 0 ) since the function is antisymmetric. The sine coefficients are:
[
b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin(n\omega_0 t) dt
]
For a standard sawtooth wave with period ( T ) and amplitude ( A ):
[
b_n = \frac{2A}{n\pi} (-1)^n
]
Thus, the Fourier series simplifies to:
[
f(t) = \frac{A}{2} – \frac{2A}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin(n\omega_0 t)
]
Step 2: Compare with Example 17.1
Example 17.1 (likely a square wave example) shows a different harmonic structure. The sawtooth wave contains only odd harmonics with alternating signs, similar to the square wave but with a different amplitude decay.
Step 3: Plot Line Spectrum
Now, I will generate the line spectrum to visually compare the harmonic components.
The plot above shows the line spectrum of the sawtooth wave, where only sine harmonics are present (odd harmonics) and their magnitudes decrease as ( 1/n ).
Explanation (300 Words)
The sawtooth wave is a periodic signal containing all integer harmonics, with its Fourier series consisting of only sine terms since it is an odd function. The coefficients decay as ( 1/n ), leading to a smoother convergence than a square wave, whose harmonics decay at ( 1/n ) but contain only odd harmonics.
In contrast to Example 17.1 (which likely discusses the square wave), the sawtooth wave includes both even and odd harmonics with alternating signs. The square wave only contains odd harmonics with amplitudes proportional to ( 1/n ), whereas the sawtooth wave includes all harmonics with a systematic amplitude decay.
The line spectrum visualizes these harmonics, with bars representing the magnitude of each harmonic component. As seen, the higher-order harmonics contribute less to the waveform, meaning that truncating the series at a reasonable ( n ) gives a good approximation of the original waveform.
This Fourier series representation is useful in signal processing, audio synthesis, and electrical engineering, where sawtooth waves are used in synthesizers, oscillators, and waveform analysis. By reconstructing the sawtooth wave with a finite number of harmonics, engineers can approximate signals efficiently while minimizing computational effort.
Overall, the sawtooth wave’s Fourier series provides insights into harmonic content, enabling applications in both theoretical and practical domains.