Spectral characteristics of DDPM streams and their application to all-digital amplitude modulation

A new closed-form expression of the spectral coefﬁcients of the digital streams obtained by dyadic digital pulse modulation is presented and validated in this letter. The new expression provides in-depth insight into the spectral properties of dyadic digital pulse modulation, revealing its applicability as an all-digital bandpass amplitude modu- lation technique. Simulations and measurements on a proof-of-concept dyadic digital pulse modulation amplitude modulator prototype demon- strate the effectiveness of the approach.

DDPM streams: The DDPM modulation [1] associates to an unsigned integer n on N bits, expressed in terms of its binary representation obtained by superposition of orthogonal dyadic basis signals (ODBSs) S i (t ) (i = 0, . . . , N − 1) defined on the basic period (0, 2 N T clk ) as: where T clk is the clock cycle and (x) is the unit pulse ( (x) = 1 for 0 ≤ x ≤ 1 and (x) = 0 elsewhere). In other words, ODBSs S i (t ) are N non-overlapping, periodically repeated digital streams of 2 N clock cycles, organized so that S N−1 is high (i.e. at V DD ) every other clock cycle (i.e. in 2 N−1 cycles per period), S N−2 is high every other cycle in which S N−1 is low (i.e. in 2 N−2 cycles), S N−3 is high every other cycle in which both S N−1 and S N−2 are low (i.e. in 2 N−3 cycles per period) and so on, till S 0 , which is high just in one cycle per period, as shown in Figure 1(a). Being ODBSs S i non-overlapping and high in 2 i clock cycles per period, DDPM streams n defined in (1) are high for exactly n clock cycles per period and their time average is therefore n/2 N V DD , as it can be observed in the same Figure 1(a), where the construction of a DDPM stream by superposition of ODBSs is illustrated for n = 10.
In practice, DDPM streams can be generated by tiny all-digital hardware (a priority multiplexer in which the data inputs are fed by the sig-nals B n [N − 1 : 0] and the selection inputs are driven by a free-running binary counter, as shown in Figure 1(b) and have been applied thus far in baseband D/A and A/D conversion and in power electronics [1] in view of their low-frequency spectral features, which make it possible to extract their DC component by a low pass filter with non-stringent requirements.
A new expression of the spectra of DDPM streams is introduced here to gain more insight into the modulation and into the potential applications of its high-frequency harmonic components.
Spectra of DDPM streams: The spectra of a DDPM stream have been evaluated in [1] by the Fourier transform of (1) as: in which δ[·] is the Kroenecker function defined as Although in closed form, Equation (3) provides a limited insight into the spectral characteristics of DDPM signals since the coefficients c k,n , which give the magnitude and the phase of the k-th spectral component of the DDPM stream n , are expressed in (4) in terms of nested summations, from which their dependence on the input code B n is not evident.

New closed-form expression of the spectral coefficients:
The expression of the DDPM spectral coefficients in (4) can be simplified considering that, for a given k = 0, the inner summation in (4) is non-zero only if k is an integer multiple of 2 i . As a consequence, being that is, the largest exponent ν such that 2 ν divides k, which is know in number theory [6] as dyadic order of k ∈ N, all the terms of the outer summation in (4) for i > ν 2 (k) give zero contribution to the spectral coefficient c k,n . Moreover, for i = ν 2 (k), the Kroenecker delta in the inner summation is non-zero just for one value of m, which is necessarily odd. In fact, if we assume, by contradiction that k = m · 2 i with i = ν 2 (k) and m even, then 2 ν2 (k)+1 would be also a divider of k and ν 2 (k) would not be the largest exponent such that 2 ν2 (k) divides k, in contrast with its definition in (5). Being m odd, the phase factor e − jπm is equal to -1 and the whole inner summation in (4) evaluates to -1 for i = ν 2 (k).
Finally, for i < ν 2 (k), the inner summation in (4) is non-zero just for one value of m, which is necessarily even. The condition i < ν 2 (k) (with strict inequality), in fact, implies that 2 i+1 also divides k, which in turn implies that if k = m · 2 i for an integer m, then m must include at least a factor 2 in its prime factor decomposition, that is, it should be even. Being m even, the phase factor e − jπm is equal to +1, and the whole inner summation in (4) evaluates to +1 for i < ν 2 (k).
Based on the above considerations, the inner summation in (4) can be explicitly evaluated and the expression of c k,n for 1 k = 0 can be written as and is very interestingly equal to the value of the binary string 2 interpreted as a signed integer in two's complement representation [7]. In other words, Equation (6) reveals that the spectral coefficient c k,n of the k-th harmonic of the spectrum of the DDPM stream for the input code n is always real, is equal for harmonics k of the same dyadic order and takes the value of the last ν 2 (k) + 1 LSBs of the binary input string B n , interpreted as a signed integer in two's complement, where ν 2 (k) is the dyadic order of the harmonic number k, as illustrated in Figure 2.
Considering the relation between the two's complement notation and the modular arithmetic [7], Equation (6) can be equivalently written as where mod is the arithmetic modulo operator. The values of the harmonic coefficients c n,k provided by Equations (6) and (7) are the same obtained directly by taking the fast Fourier transform (FFT) of DDPM streams, which are reported for N = 4 in Figure 3 for direct validation of (6) and (7).
Application to all-digital amplitude modulation: From an application perspective, Equation (6) reveals that the amplitude of the harmonics of DDPM streams with dyadic order ν 2 (k) can be varied over 2 ν2 (k)+1 uniformly spaced levels by setting the last ν 2 (k) + 1 LSBs of B n . In particular, by varying the digital input B n of a DDPM modulator according to a sequenceñ[h] of N-bit signed integers in two's complement, as illustrated in Figure 4(a), the amplitude of the spectral line at frequency 2 N−1 f 0 = f clk /2 of the output stream -which can be regarded as a carrier -can be modulated at N-bit resolution according to the digital modulating sequenceñ[h] over a 1/(2T 0 ) = f clk /2 N+1 modulation bandwidth, dictated by the duration of the DDPM pattern T 0 = 2 N T clk .
Simulations: The equivalence highlighted in Figure 4 is confirmed by the simulation results presented in Figure 5, where the spectra of an 8bit DDPM stream (Figure 4(a), before the bandpass filter [BPF]) and of a BP-PAM signal (Figure 4(b), before the BPF), both modulated at f clk /2 8 Fig. 4 Narrowband equivalence around f clk /2, revealed by Equation (6) where · is the rounding to the nearest integer operator, are compared around f clk /2 on three different scales, revealing near identical spectral content in the modulation sidebands.
From Figure 5, it can be observed that the amplitude of the spurious spectral components of the DDPM-modulated stream increase gradually outside the modulation bandwidth and can be therefore suppressed by a BPF with non-stringent requirements, in analogy to what happens in baseband DDPM D/A conversion [1]. This can be more rigorously demonstrated adapting the analysis under dynamic conditions presented in [1] to (6) and is also observed in Figure 6, where the time-domain   Figure 5, both filtered by a second-order BPF with Q = 40, are compared.

Measurements:
The possibility to apply DDPM as a bandpass amplitude modulation technique has been experimentally verified by measuring the output spectrum of a proof-of-concept N = 5 bit DDPM modulator synthesized on an Altera Cyclone IV FPGA in a DE-2 115 board.
The DDPM modulator operates at f clk =1 MHz and is fed by the samples of a digital sine wave at 95% full-swing amplitude and frequency f = 7/64 · f 0 = 7/64 · f clk /2 5 3.4 kHz quantized on N = 5 bits. The measured spectrum around 3 f clk /2 =500 kHz of the DDPM stream is reported in Figure 7 and is in good agreement with the simulated results under the same test conditions (reported in the same figure) and with (6), thus confirming the validity of the analysis and the applicability of DDPM as an all-digital amplitude modulation technique.

Conclusion:
A new simple, closed-form expression of the spectral coefficients of DDPM streams, which highlights a simple relation between the amplitude of the harmonics of DDPM streams and the binary representation of the DDPM modulator input, has been presented in this letter. The insight gained by the proposed analysis reveals the applicability of DDPM as an all-digital, bandpass amplitude modulation technique, which has been confirmed by simulations and experimental results on a proof-of-concept prototype.