# An Energy-Efficient Integration of a Digital Modulator and a Class-D Amplifier
## 3 Theoretical Background  
PW-modulations can be classified according to several criteria. The first is pulse values, which can be symmetric or asymmetric. The former means \(V_{L}=0\) V and the later means \(V_{L}=-V_{H}\), where \(V_{L}\) and \(V_{H}\) are, respectively, the low and high levels of the output voltage. The symmetric case is slightly more complex to generate from the circuit point of view, and its frequency spectrum has a much lower DC component and larger fundamental and harmonic components. However, other than that, it has little influence on the frequency spectrum of the modulated signal. A second criterion is related to the fact that the input is a continuous signal but the output signal is a train of pulses, which means that the input signal has been somehow sampled at certain instants. The way sampling is performed also has an impact on the frequency spectrum. There are two options, the so-called uniform sampling, when it is synchronous with the system clock, or the natural sampling, which happens when simple circuits are used to modulate the pulses, i.e., when sampling is achieved by a simple comparator circuit and is not synchronous with the system clock. The basic block diagram of a PW-modulator is shown in Figure 1[8]. Of course, the average sampling frequency over a long period of time is the frequency of the system clock. The third criterion is related to how the sample modulates the output pulses. The samples can determine the positions of either the leading edges, the trailing edges, or both. In the double-edge case, the position of both edges can be symmetrically determined by a single sample or each of them can be determined by two consecutive samples. [PERSON] and [PERSON] determined the frequency spectra of different possible PW-modulations [9]. A brief summary of their work is presented here, for the sake of completeness. Let us assume the following conditions: (i) Symmetric pulse amplitudes are \(-1\) and \(1\) V. (ii) The input signal \(x(t)\) is frequency-limited, the module of its maximum amplitude is \(1\) V, and it has no DC component. (iii) The modulation index is 100% and the idle duty cycle is 50%. (iv) The pulse repetition frequency \(f_{c}\) (better than sampling frequency in natural sampling) is at least \(\pi\) times the maximum frequency of the input signal. Under these conditions, the Fourier expansions of the natural sampling of the PW-modulated signals are [9]:  
\[p_{\mathrm{TE}}(t)=x(t)+\sum_{k=1}^{\infty}\frac{2}{k\pi}\left[\sin(2\pi kf_{ c}t)-(-1)^{k}\sin(2\pi kf_{c}t-k\pi x(t))\right], \tag{1}\]  
Figure 1: Basic block diagram of a PW-modulator [8]. \[p_{\rm LE}(t)=x(t)+\sum_{k=1}^{\infty}\frac{2}{k\pi}\Big{[}(-1)^{k}\sin(2\pi kf_{ c}t+k\pi x(t))-\sin(2\pi kf_{c}t)\Big{]}\,, \tag{2}\] \[p_{\rm DE}(t)=x(t)+\sum_{k=1}^{\infty}\frac{2(-1)^{k}}{k\pi}\Big{[} \sin\!\left(2\pi kf_{c}t+k\pi\frac{x(t)+1}{2}\right)-\sin\!\left(2\pi kf_{c}t- k\pi\frac{x(t)+1}{2}\right)\!\Big{]}, \tag{3}\]  
where the subscripts TE, LE, and DE stand, respectively, for the trailing-edge, leading-edge, and double-edge (asymmetric version).