In accordance with the methods and systems disclosed herein, non-linear functions may be added within a loop to accomplish two “different” objectives, namely: (1) to achieve higher-order shaping of the non-linearity of the multi-level PWM pulse, despite sampling after only two continuous integrators; and (2) to correct for errors in the impulse response caused by sampling the first continuous integrator “before” the end of the PWM frame.

Achieving Higher Order Shaping

To aid in illustration of the impact of locating early sampling at lower-order continuous-time integrators 38, mode M=0 and duty cycle D=0.5 are chosen for illustrative purposes. However, the following discussion may apply for any arbitrary mode M and duty cycle D.

FIG. 7 illustrates example waveforms for sampling and processing of a signal through modulator 28A depicted in FIG. 5, in accordance with embodiments of the present disclosure. In particular, FIG. 7 illustrates a signal value at the outputs INT1_out, INT2_out, and INT3_out of each of continuous-time integrators 38 of modulator 28A. For example, as shown in FIG. 7, if outputs are sampled at the mid-point of a PWM frame, the values of INT1_out, INT2_out, and INT3_out may be D/2, D²/8, and D³/48, respectively. If outputs are sampled at the end of a PWM frame, the values of INT1_out, INT2_out, and INT3_out may be D, D/2, and D/8+D³/24, respectively.

In the example of FIG. 7, the sampled values at the midpoint and end of each sample period are based on a positive duty cycle at mode M=0. In general, for any linear combination of M+D (wherein D may be negative), the result from M may be treated as D=1 with scalar M. Therefore, the endpoint values from the mode pulse for INT1_out, INT2_out, and INT3_out may be M, M/2, and M/8+M/24, respectively. Results of a negative duty cycle D may be similar to that shown in FIG. 7, with the small difference that when duty cycle D is negative, all square terms will equal sign(D)·D².

When all of the derived equations are applied to two scenarios, namely, (M=0, D=0.5) and (M=1, D=−0.5), the values at the end of a sample period may be as shown in the table below.

INT1_OUT
INT2_OUT
INT3_OUT

M = 0, D = 0.5
M + D = 0.5
(M + D)/2 =
(M + D)/8 + D³/24 +

0.5/2
M/24 = 0.5/8 + 0.5³/24

M = 1, D = −0.5
M + D = 0.5
(M + D)/2 =
(M + D)/8 + D³/24 +

0.5/2
M/24 = 0.5/8 −

0.5³/24 + 1/24

As seen from the above table, for any arbitrary M and D, the sampled output of the third continuous integrator (INT3_OUT), when sampled at the end of the frame, has a non-linear term of D³/24+M/24. Thus, with respect to sampling of the outputs of integrators 38 in the embodiments represented by FIG. 5, a considerable non-linear discontinuity of the third integrator output INT3_OUT in the amount 1/32 occurs when transitioning from mode M=0 to mode M=1, as shown in FIG. 8. Notably, however, the sampled output of the cascade of the first two continuous integrators, when sampled at the end of a PWM frame, includes only a linear term in M+D, and there is no discontinuity in the value of M+D.

Accordingly, sampling the second continuous integrator at a frame boundary may not capture this mode transition-induced nonlinearity. As an example, FIG. 6 depicts a system (e.g., modulator 28B) with two continuous integrators 38. The second integrator 38 may be sampled and then fed into subsequent (discrete) integrators, which may be implemented in the digital domain. These discrete integrators may be incorporated within digital loop filter 31 shown in FIG. 6. As already discussed above, when sampled at the boundary of the PWM frame, the sampled output of the second continuous integrator (INT2_OUT) may only consist of linear terms. Thus, any subsequent discrete integrator that receives the frame-boundary samples of INT2_OUT may not be able to observe the mode-transition discontinuity and may hence not be able to correct for such discontinuity.

Consequently, when sampling the second integrator 38 at the boundary of the PWM frame as shown in the embodiments represented by FIG. 6, the non-linearity caused by this abrupt change in the shape of multi-level PWM pulse would only be second-order shaped since it would not benefit from the feedback loop's third integrator. In such a scenario, the harmonics of the mode transition that appear in the output signal spectrum may cause a notable degradation in linearity that may impact all signal levels. As described in greater detail below, to minimize performance degradation due to having only two continuous integrators 38, embodiments of the present disclosure may include a non-linear function added at the input of the digital portion of the loop filter, in order to generate the same non-linear term as that present in the higher-order continuous time system to maintain higher-order noise shaping of the non-linear feedback pulse. The addition of the non-linear function may aid in minimizing in-band distortion observed at the output of speaker 24.

The systems and methods described herein may also address a correction for the error caused by early-sampling of the first continuous-time integrator 38, that is, the error caused by sampling the first continuous integrator (e.g., INT1_OUT in FIG. 7) “before” the end of the PWM frame. Although such errors in INT1_OUT may not significantly impact the in-band linearity of modulator 28B, they do impact the overall system stability and the system's transient response, especially during mode transitions.

Both the in-band distortion issue, as well as the error caused by the early sampling of the first integrator, can be addressed by adding various non-linear correction terms within the digital portion of the loop filter. Described below are three approaches for implementing an addition of a non-linear function within a digital portion of a loop filter. In the first approach, an explicit digital correction term may be added before the feedback loop's third (discrete-time) integrator. In the second approach, the feedback loop's second (continuous-time) integrator may be sampled numerous times, with a combination of these samples received by the feedback loop's third (discrete-time) integrator. In the third approach, a non-linear function is added to address early sampling of the feedback loop's first (continuous-time) integrator. The first two approaches may assist in achieving higher-order shaping of the feedback pulse's non-linearity. The third approach may assist in enhancing the system's stability and high-frequency transient response by restoring the impulse response of the first order path.

FIG. 9 illustrates an output bridge driver 22 and selected components of a modulator 28C having a second-order continuous-time analog loop filter with early sampling and a post-sampling higher-order digital loop filter 31 having a non-linear correction function 40 added thereto, in accordance with embodiments of the present disclosure. Modulator 28C depicted in FIG. 9 may be similar in many respects to modulator 28B of FIG. 6, and thus, only certain differences between modulator 28C and modulator 28B may be described below.

For instance, modulator 28C may include non-linear correction function 40 which may be applied to the output of PWM modulator 36 with the result added to the output of sampler 32 before being received by a higher-order integrator of digital loop filter 31. As explained above, when sampled at the end of a PWM frame, the sampled output of the second continuous-time integrator 38 may include only terms that are linear with respect to the output of quantizer 34. Thus, to cause the output of the feedback loop's third (discrete time) integrator, present within digital loop filter 31, to be equivalent to what the sampled output of the third integrator would have been if implemented as an analog continuous-time integrator, non-linear correction function 40 may apply a correction term c[n] to the output of PWM modulator 36 and added to the input of the feedback loop's third integrator. In accordance with the examples described above with respect to FIGS. 5 and 6, such correction term may be c[n]=D³/24+M/24. As a result, addition of non-linear correction term c[n] to the input of a third integrator of a feedback loop, wherein the third integrator is implemented as a discrete-time digital integrator, may achieve third-order shaping of the non-linearity from the multi-level PWM pulse.

FIG. 10 illustrates an output bridge driver 22 and selected components of a modulator 28D having a second-order continuous-time analog loop filter with early sampling and a post-sampling higher-order digital loop filter 31, wherein correction is applied on the basis of multiple sample points of the output of second continuous-time integrator 38, in accordance with embodiments of the present disclosure. Modulator 28D depicted in FIG. 10 may be similar in many respects to modulator 28C of FIG. 9, and thus, only certain differences between modulator 28D and modulator 28C may be described below.

In modulator 28D, sampler 32A (used in lieu of sampler 32 of modulator 28C) may take multiple samples of the output of the second integrator 38 during a single PWM frame in order to implement a non-linear function. Referring again to FIG. 7, it is seen that when sampled at the midpoint of a PWM frame, integrator output signal INT2_OUT includes a term sign(D)·D²/8+M/8, whose time-domain behavior across mode transition boundaries is quite similar to that of c[n]=D³/24+M/24. For example, a mode transition-induced discontinuity in sign(D)·D²/8+M/8 may be 2/32, which is twice the mode transition-induced continuity in D³/24+M/24. Thus, if the input of the feedback loop's third integrator (implemented digitally within digital loop filter 31) is set to be a particular combination of the mid-frame sample (at time T1) and end-frame sample (at time T2) of integrator output signal INT2_OUT, then, even without adding any explicit correction term, the mode transition-induced discontinuity added in modulator 28D may be matched to c[n]=D³/24+M/24. Thus, inputting into the third integrator a weighted combination (as generated by a weighting function 42) of the mid-frame sample (at time T1) and end-frame sample (at time T2) of integrator output signal INT2_OUT may provide non-linearity correction that approximates that of modulator 28C of FIG. 9.

Correcting for Errors in the Impulse Response

FIG. 11 illustrates an output bridge driver 22 and selected components of a modulator 28E having a second-order continuous-time analog loop filter with early sampling and a post-sampling higher-order digital loop filter 31, wherein a non-linear function is applied to compensate for early sampling of the first integrator of the feedback loop, in accordance with embodiments of the present disclosure. Modulator 28E depicted in FIG. 11 may be similar in many respects to modulator 28C of FIG. 9, and thus, only certain differences between modulator 28E and modulator 28C may be described below.

For example, as shown in FIG. 11, modulator 28E may, in lieu of non-linear correction function 40, apply a non-linear correction function 50 with a correction term r[n] to a feedback signal output by PWM modulator 36 which is based at least in part on the sampled output of the first integrator 38 by sampler 32. To illustrate, because the output of the first continuous-time integrator 38 may be sampled before the end of a PWM frame (e.g., at a time instant t=T0), the output INT1_OUT of the first integrator 38 may have an error which is a function of mode M and duty cycle D. The nature of the error may be different between the mode pulse and the PWM pulse. When the mode pulse is processed by the first continuous-time integrator 38 and then sampled at time t=T0, the sampled output INT1_OUT may have an error which is proportional to the shaded area shown in FIG. 12.

As shown in FIG. 12, at time t=T0, the output of the first continuous-time integrator 38 has not yet reached its final value. Thus, sampling the output of the first continuous integrator at time t=T0 introduces an error E=M(T−T0). To compensate for this error, a correction term equal to M(T−T0) must be added to the sampled output of first continuous-time integrator 38. The nature of this error is plotted in FIG. 13. For simplicity, the subsequent analyses will assume T=1 and that T0 is a fraction which is normalized to T. Under this assumption, the correction for the mode can be re-written as M(1−T0).

To correct for the PWM pulse, the pulse width |D| of the PWM pulse may be determined to have either a first condition in which |D|<(2T0−1) or a second condition in which |D|>(2T0−1). In the first condition, the output INT1_OUT of the first continuous-time integrator may reach its final value before the sampling instant T0 as shown in FIG. 14A. Thus, in this first condition, no correction may be required. However, in the second condition, the output INT1_OUT of the first continuous-time integrator may reach its final value after the sampling instant T0 as shown in FIG. 14B. Thus, in the second condition, an error E=0.5[D−(2T0−1)] may be introduced into the impulse response sample at n=1. Such error E may be corrected by applying a non-linear correction sequence defined by −0.5[D−(2T0−1)] 6[n−1]. Similarly, when duty cycle D<−(2T0−1), such error may be corrected by applying a non-linear correction sequence defined by −0.5[D+(2T0−1)] 6[n−1].