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Application of the Fourier Series to Filters

Filters are an important component of electronics and communications systems. Chapter 21 presented a full discussion on passive and active filters. Here, we investigate how to design filters to select the fundamental component (or any desired harmonic) of the input signal and reject other harmonics. This filtering process cannot be accomplished without the Fourier series expansion of the input signal. For the purpose of illustration, we will consider two cases, a lowpass filter and a bandpass filter. The output of a lowpass filter depends on the input signal, the transfer function $ H(\omega) $ of the filter, and the

Application of the Fourier Series to Spectrum Analyzers

The Fourier series provides the spectrum of a signal. As we have seen, the spectrum consists of the amplitudes and phases of the harmonics versus frequency. By providing the spectrum of a signal $ f(t) $, the Fourier series helps us identify the pertinent features of the signal.
It demonstrates which frequencies are playing an important role in the shape of the output and which ones are not. For example, audible sounds have significant components in the frequency range of $ 20 \mathrm{~Hz} $ to $ 15 \mathrm{kHz} $, while visible light signals range from $ 10^{5} \mathrm{GHz}

Exponential Fourier Series

A compact way of expressing the Fourier series in Eq. (A) is to put it in exponential form. $$f(t)=a_{0}+\sum_{n=1}^{\infty}\left(a_{n} \cos n \omega_{0} t+b_{n} \sin n \omega_{0} t\right) \tag{A}$$ This requires that we represent the sine and cosine functions in the exponential form using Euler's identity: $$\begin{aligned}\cos n \omega_{0} t &=\frac{1}{2}\left[e^{j n \omega_{0} t}+e^{-j n \omega_{0} t}\right] \\ \sin n \omega_{0} t &=\frac{1}{2 j}\left[e^{j n \omega_{0} t}-e^{-j n \omega_{0} t}\right]\end{aligned} \tag{1}$$ Substituting Eq. (1) into Eq. (A) and collecting terms, we obtain $$f(t)=a_{0}+\frac{1}{2} \sum_{n=1}^{\infty}\left[\left(a_{n}-j b_{n}\right) e^{j n \omega_{0} t}+\left(a_{n}+j b_{n}\right) e^{-j n \omega_{0} t}\right] \tag{2}$$ If we define a new coefficient $ c_{n} $ so that $$c_{0}=a_{0}, \quad

Average Power and RMS Values of The Fourier Series

To find the average power absorbed by a circuit due to a periodic excitation, we write the voltage and current in amplitude-phase form as $$v(t)=V_{\mathrm{dc}}+\sum_{n=1}^{\infty} V_{n} \cos \left(n \omega_{0} t-\theta_{n}\right) \tag{1}$$ $$ i(t)=I_{\mathrm{dc}}+\sum_{m=1}^{\infty} I_{m} \cos \left(m \omega_{0} t-\phi_{m}\right) \tag{2}$$ Following the passive sign convention (Fig. 1), the average power is $$P=\frac{1}{T} \int_{0}^{T} \text { vi } d t \tag{3}$$ Substituting Eqs. (1) and (1) into Eq. (3) gives $$\begin{aligned}P=& \frac{1}{T} \int_{0}^{T} V_{\mathrm{dc}} I_{\mathrm{dc}} d t+\sum_{m=1}^{\infty} \frac{I_{m} V_{\mathrm{dc}}}{T} \int_{0}^{T} \cos \left(m \omega_{0} t-\phi_{m}\right) d t \\&+\sum_{n=1}^{\infty} \frac{V_{n} I_{\mathrm{dc}}}{T} \int_{0}^{T} \cos \left(n \omega_{0} t-\theta_{n}\right) d t \\&+\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{V_{n} I_{m}}{T} \int_{0}^{T} \cos \left(n \omega_{0} t-\theta_{n}\right) \cos \left(m \omega_{0}

Circuit Application of The Fourier Series

We find that in practice, many circuits are driven by nonsinusoidal periodic functions. To find the steady-state response of a circuit to a nonsinusoidal periodic excitation requires the application of a Fourier series, ac phasor analysis, and the superposition principle. The procedure usually involves three steps. Steps for Applying Fourier Series: Express the excitation as a Fourier series. Find the response of each term in the Fourier series. Add the individual responses using the superposition principle. The first step is to determine the Fourier series expansion of the excitation. For the periodic voltage source shown in Fig. 1(a), for example, the Fourier series is expressed
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