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Texas Instruments Audio and Video/Imaging Series
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Click modulation is a pulse position modulation, from which you can easily derive a binary signal. The spectral content of this binary signal includes the source's spectrum and the modulation products, but the latter does not overlap with the former, as is customary in PWM techniques [4]. This article presents a brief introduction to the click modulation technique. The mathematical details are presented to a bare necessity; the interested reader should consult [2].
Figure 1: Block diagram of the click modulator |
The scheme of a click modulator is shown in Figure 1. Given a bandlimited, passband-like signal f(t) with spectral content confined to [-
H, -L] ∪ [L, H], as shown in Figure 2a, where 0 < L < H < 
, and L is strictly greater than zero (in other words, f(t) has a zero DC component) an analytic signal fA(t) is derived as:
(1)
where
(2)
is the Hilbert transform of f(t) (the symbol "*" represents convolution in time domain). The analytic signal fA(t), whose spectral content is limited to [L, H], is fed through an analytic exponential modulator (AEM) to yield:
(3)
The output z(t) = x(t) + jy(t) of the AEM is also analytic, but its spectral content covers the entire positive frequency axis, as shown in Figure 2c.
Figure 2: Spectra of diferent signals of the click modulator |
Under the restriction |f(t)| < π/2, the analytic extension z(τ), τ ∈ C of z(t) is free of zeros in Im{τ} > 0, and therefore, {-j log[z(t)]} coincides with f(t). The output z(t) of the AEM is bandlimited by a low-pass filter with impulse response hA(t) and corner frequency S, such that it completely blocks all signals of frequency greater than U, where 0 < L < H < S < U < , to yield:
The filter output zf(t) = xf(t) + j yf(t) (Figure 2d) is still analytic, and if |zf(t)| < π/2, its analytic extension z(τ) is also free of zeros in Im{τ} > 0. Therefore, the Fourier transform of fA(t) = j log z(t) and that of j log zf(t) (Figure 2e) coincide in the frequency range (-, S] (assuming that L > 0).
The signal s(t), defined as
(4)
with P > (S + U)/2, has real and simple zeros, and can be uniquely defined by their position (in other words, by its zero crossings). The spectrum of signal s(t) is limited to [-P, P], and resembling a lower sideband with carrier, as depicted in (Figure 2f). Therefore, the frequency spectrum of
(5)
is confined to the interval [0, 2P] (Figure 2g). More precisely, it coincides with the spectrum of zf(t) (Figure 2d) within the range (-, S], and therefore, the spectra of j log u(t), j log zf(t) and of fA(t) are equal over the interval (-, S], as shown in Figures 2h, 2e, and 2b, respectively. In consequence, the Fourier transform of the signal
(6)
(where the principal branch of the log function is taken) is exactly the same of the Fourier transform of the original f(t) signal in the baseband [-S, S] (see Figures 2i and 2j), and has high-frequency components that are isolated from the baseband by the frequency interval [H, S] (guardband). Therefore, signal f(t) can be recovered by a simple low-pass filtering of v(t) with a filter HR(s) with unity-gain in the band [-H, H] and rejecting signals of frequency greater than S (Figure 2k)
(7)
as shown in Figure 2l, where hR(t) is the impulse response of the reconstruction filter.
Coding with Binary Signals
A binary signal with the same spectral content as v(t) can be easily derived. Figure 3 shows some of the signals illustrated in Figure 1. From the source signal f(t) (Figure 3a), you can obtain signal s(t) (Figure 3b) according to the previous section. Noting that
it follows from Equations 5 and 6 that v(t) = -arg{s(t)} - Pt, and according to [2], -π < v(t) + arg{s(t)} < π (Figure 3c). It should be noted that s(t) is a real signal (see Equation 4). Also, arg{s(t)} = (π/2)(sgn[s(t)] - 1) varies between 0 and ±π whenever s(t) changes polarity, and therefore, -arg{s(t)} - Pt is a negative ramp with jumps of ±π. This reveals that the zeros of v(t) occur due to both the sign change of s(t) and the jump from (-π) to (+π) caused by the ramp (-Pt) on the principal branch of the complex log function. According to the frequency relationship between s(t) and ejPt, the zeros of s(t) are located between the zeros of sin(Pt). The phase-change generated by the ramp (-Pt) occurs at every 2π/(2P) seconds, and produces distortion at multiples of 2P, that laid outside of the frequency band of interest.
Figure 3: Different waveforms of the click modulator |
The characteristic of v(t) suggest a way to generate a binary signal q(t) exhibiting the same frequency content as v(t). A periodic sawtooth σ(t) (Figure 3d) with repetition period π/P
such that σ(t) = σ(t + π/P), has zero spectral content within the band [-2P, 2P], as you can see by computing its Fourier series.
The zeros of σ(t) occur at the same time as those of the zeros of sin(Pt). By subtracting σ(t) from v(t), we obtain a variable duty cycle square wave signal q(t) (Figure 3e):
(8)
This bipolar signal is completely determined by the position of the zeros of both s(t) (Figure 3f) and of sin(Pt) (Figure 3g). From the definition of q(t) and the spectral characteristic of v(t) and σ(t), the spectral of q(t) and v(t) are identical in the frequency interval [-S, S], and, as noted previously, σ(t) has zero spectral content in the band [-2P, 2P]. As such, the spectral of f(t) and q(t) coincide over the interval [-S, S], as shown in Figure 4, and therefore, f(t) can be recovered by low-pass filtering of q(t) with a reconstruction filter HR(s) having a unity-gain bandwidth of H and blocking any frequency component above S
Equation 8 shows how you can obtain q(t) from s(t).
Figure 4: Spectrum of q(t) showing the isolation between the baseband and the high frequency distortion components |
PWM vs. Click Modulation
Click modulation forces transitions in the binary signal q(t) from low to high at the zeros of s(t), in other words, when cos(arg{s(t)} - pt) = 0 (Equation 4). The roots of this equation occur when
(9)
Remembering that σ(t) is the sawtooth shown in Figure 3d, click modulation exhibits leading-edge transitions when Equation 9 is satisfied, and trailing edge transitions periodically, at a rate of π/p whenever sin(pt) changes sign. If the filter HA(s) after the analytic exponential modulator is omitted (and if |f(t)| < π/2), it happens that arg{s(t)} = -f(t), and therefore, Equation 9 compares the data signal f(t) with the sawtooth σ(t). In this way, the modulator (without filter HA(s)) produces a natural-sampled PWM signal [12], as shown in Figure 5.
Figure 5: Natural PWM scheme [12] or phase modulation for SSB [1] |
Click modulation can be thought as a modified PWM process [12] or a modified phase-modulation for SSB [1]. The passband filter HA(s) avoids the appearance of high-frequency intermodulation products in the signal baseband. Comparatively, to achieve similar distortion levels, PWM techniques requires a much higher switching frequency, in the range of 250 - 500 kHz in some commercial amplifiers.
However, the complexity of the click modulation algorithm precludes an analog implementation, and a discrete-time implementation is highly desired. This introduces additional deleterious effects that should be carefully addressed.
First, you should design the (discrete) analytical exponential modulator (AEM) with great care, to diminish the adverse aliasing that can occur. The output of the continuous-time AEM is not bandlimited, and therefore, a direct discrete-time translation is prone to fail due to aliasing. Two alternatives can be explored:
- To use signal processing techniques such as oversampling and interpolation, to narrow the signal bandwidth and ameliorate the adverse alias effects
- To approximate the complex exponential function by a low-order complex polynomial. In this case, you can easily compute the maximum frequency content of the output of the (approximate) AEM, and a proper sampling frequency can be chosen to avoid aliasing at all.
Second, you should compute the zero-crossing of s(t) with a very high resolution, because failing to do so would increase the noise level. This is a serious drawback of the hardware implementations, demanding high processing speeds because these issues should be resolved in real time.
Finally, the click modulation cannot process low pass-signals, only band-pass signals are allowed, to guarantee that the signal changes sign. Furthermore, digital Hilbert-transformers are easily implemented also for band-pass signals. However, this is not a serious issue, because the frequency band comprised between 0 Hz and the lower frequency corner L (if present) can be covered using conventional PWM techniques: the narrow bandwidth (with respect to the switching frequency) assures that the distortion introduced by the PWM is small.
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