CN110261678B - Left-increasing double-edge UPWM signal frequency spectrum estimation method based on digital PWM generator - Google Patents

Left-increasing double-edge UPWM signal frequency spectrum estimation method based on digital PWM generator Download PDF

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CN110261678B
CN110261678B CN201910693395.0A CN201910693395A CN110261678B CN 110261678 B CN110261678 B CN 110261678B CN 201910693395 A CN201910693395 A CN 201910693395A CN 110261678 B CN110261678 B CN 110261678B
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于泽琦
陈光伟
张恩光
田二林
王凤琴
张珂
陈晓雷
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Zhengzhou University of Light Industry
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    • G01—MEASURING; TESTING
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Abstract

The invention provides a left-increasing double-edge UPWM signal frequency spectrum estimation method based on a digital PWM generator. And then, carrying out discrete Fourier transform on the discrete points by using the distribution rule of the obtained discrete point amplitudes, and multiplying the transform result by the frequency response function of the zero-order retainer to obtain a left-increased double-edge two-level UPWM signal frequency spectrum estimation expression. And subtracting the two-stage UPWM signal frequency spectrum estimation expressions obtained by modulating the input signal of the digital PWM generator before and after inversion to obtain the corresponding three-stage UPWM signal frequency spectrum estimation expressions. The method provided by the invention can directly obtain the frequency spectrum of the left-increased double-edge UPWM signal under the condition that the input modulation signal is a digital signal.

Description

Left-increasing double-edge UPWM signal frequency spectrum estimation method based on digital PWM generator
Technical Field
The invention relates to the field of digital D-type audio power amplification, in particular to a frequency spectrum estimation method of a left-increased double-edge uniform sampling pulse width modulation signal obtained by modulating a digital audio signal.
Background
The power efficiency of the digital D class audio power amplifier is higher than that of the linear audio power amplifiers such as A class, B class and AB class, and the digital D class audio power amplifier also has the advantage of being capable of directly interfacing with a digital audio source, so that the digital D class audio power amplifier becomes a research hotspot in the field of power amplifiers. A schematic structure diagram of a digital class D audio power amplifier is shown in fig. 1, and generally includes a digital interpolation filter, a Sigma-Delta modulator, a digital Pulse Width Modulation (PWM) generator, a power stage, and an analog low-pass filter. In fig. 1, a digital interpolation filter oversamples a power amplifier input digital Signal, a Sigma-Delta modulator performs requantization and quantization Noise shaping on the oversampled Signal, converts a high-precision Signal into a low-precision Signal, and simultaneously transfers quantization Noise to a high frequency to maintain a Signal-to-Noise Ratio (SNR) in an input Signal baseband; an output signal of the Sigma-Delta modulator is modulated into a Uniform sampling Pulse Width Modulation (UPWM) signal by a digital PWM generator; the UPWM signal is amplified by a power stage and is finally restored into an analog signal by an analog low-pass filter so as to drive a loudspeaker to produce sound.
The digital PWM generator modulates the input digital modulation signal into a UPWM signal using UPWM techniques. Since the UPWM is a non-linear modulation technique, the input digital modulation signal generates non-linear amplitude and phase distortions of the modulation signal and intermodulation distortion of the modulation signal and the carrier signal when passing through the digital PWM generator. In order to design a high-performance digital class-D audio power amplifier, it is necessary to evaluate the distortion of the UPWM signal. Because the UPWM signal in the time domain cannot be directly analyzed to evaluate the distortion condition of the modulation signal in the modulation process, the UPWM signal can be converted into the frequency domain to obtain the frequency spectrum of the UPWM signal, and the distortion condition of the modulation signal can be analyzed and researched through frequency spectrum analysis.
The method currently used for estimating the frequency spectrum of UPWM signals is mainly a dual fourier series method (Nielsen K.A review and composition of Pulse Width Modulation (PWM) methods for estimating and digital input switching power amplifiers [ C ]// processes of the 102 and AES Convention, Munich, Germany, Audio Engineering Society,1997, Preprint 4446.) which gives an accurate spectral expression of UPWM signals obtained when the modulation signal is a single-frequency analog signal or a discrete-time signal, with the disadvantage that the method is no longer applicable when the modulation signal is a multi-frequency signal or a digital signal. Song et al first gives an approximate expression of the UPWM Signal in the time domain using its pulse width in relation to the carrier Signal in the time domain, and then obtains a spectral expression of the UPWM Signal by computing the Fourier transform of the expression (Song Z, Sardate D V.the frequency spectrum of pulse width modulated signals [ J ] Signal Processing,2003,83(10): 2227-. The method is also applicable when the modulated signal is a multi-frequency signal. When the modulation signal is a single-frequency signal, the result obtained by the method is the same as the result obtained by the double Fourier series method. But this method is equally not applicable to the case when the modulated signal is a digital signal. According to the analysis, when the modulation signal is a digital signal, no relevant literature provides an accurate UPWM signal spectrum estimation expression.
Disclosure of Invention
The invention provides a left-increasing double-edge UPWM signal frequency spectrum estimation method based on a digital PWM generator, which is suitable for the condition of inputting digital signals.
In order to solve the technical problems, the invention adopts the following technical scheme: a left-increasing double-edge UPWM signal frequency spectrum estimation method based on a digital PWM generator comprises the steps of firstly discretizing and quantizing a left-increasing double-edge two-stage UPWM signal output by the digital PWM generator on a time domain according to sampling frequency and quantization grade of the digital signal input by the digital PWM generator to obtain a series of discrete points; then, discrete Fourier transform is carried out on the discrete points by utilizing the distribution rule of the obtained discrete point amplitudes, and the transform result is multiplied by the frequency response function of the zero-order retainer to obtain a left-increased double-edge two-level UPWM signal frequency spectrum estimation expression; and subtracting the two-stage UPWM signal frequency spectrum estimation expressions obtained by modulating the input signal of the digital PWM generator before and after inversion to obtain the corresponding three-stage UPWM signal frequency spectrum estimation expressions.
The method comprises the following steps:
the first step is as follows: for reference analog signal s (t) at sampling frequency fsSampling and quantizing to obtain digital signal sq[m]T belongs to R, m belongs to Z, R represents real number, Z represents integer, k is used in quantization processqRepresenting the quantization level, kqEqual to the number of steps obtained by equally dividing the amplitude of s (t) into equal steps plus 1, and quantizing the digital signal sq[m]∈[0,kq-1]And s isq[m]∈Z；
The second step is that: the digital PWM generator converts the digital signal sq[m]Modulating into a left-increased double-edge two-level UPWM signal recorded as yGLDAD(t)；
The third step: will increase the left double-edged two-level UPWM signal yGLDADEach pulse period of (t) is equally divided in the time domain into kpwmPart, kpwmThe value of (A) usually satisfies the exponential form of 2, and kpwm＝kq-1，kpwmA number of stages, referred to as a digital PWM generator, also referred to as pulse width resolution; then for yGLDAD(t) discretization in the time domain, yGLDAD(t) each discrete point is distributed over the start time of each signal after division; then, carrying out quantization processing on the discrete points, wherein when the amplitude is high level, the value of the discrete point is 1, and when the amplitude is low level, the value of the discrete point is 0; corresponding to pair yGLDAD(t) at a sampling frequency fcSampling and quantizing, fc＝kpwm·fs(ii) a These discrete points can form a digital signal sequence, denoted as y, which approximates the original left-growing double-edge two-level UPWM signalGLDAD[n]N belongs to Z, N is more than or equal to 0 and less than or equal to N-1, and N is the total number of the discrete points; if a digital signal s is inputq[m]When the length of (A) is M, N is M.kpwm；
The fourth step: for yGLDAD[n]The distribution rule of discrete point amplitude is analyzed and researched, and the following results can be obtained:
Figure GDA0002955991880000031
wherein the symbol (#)upAnd ()downBy rounding up and down "", respectively, e.g., (5/2)up＝3，(5/2)down2; for digital signal yGLDAD[n]Performing discrete Fourier transform, and then combining the obtained Fourier transform expression with the frequency response function h [ k ] of the zero-order keeper]Multiplying, k is more than or equal to 0 and less than or equal to N-1, k belongs to Z, and obtaining a digital modulation signal sq[m]And the frequency spectrum estimation expression of the modulated left-increased double-edge two-stage UPWM signal is recorded as yGLDAD[k](ii) a The frequency response function of the zeroth order keeper is generally expressed as:
Figure GDA0002955991880000041
wherein j is a unit of an imaginary number,
Figure GDA0002955991880000042
Tcis yGLDAD[n]Sampling period of (D), Tc＝1/fc＝1/(kpwm·fs)，Ω＝2πkfcN; simplifying h (j Ω) to obtain:
Figure GDA0002955991880000043
the spectral estimation expression of the left-growing double-edge two-level UPWM signal is:
Figure GDA0002955991880000044
formula (4) is simplified to obtain:
Figure GDA0002955991880000045
the fifth step: in order to obtain a frequency spectrum estimation expression of a left-increased double-edge three-level UPWM signal, an input digital signal s is firstly inputq[m]Taking the inverse to obtain a digital signal sq_n[m]The inverse process is sq_n[m]＝kpwm-sq[m]At this time, the digital signal s in the formula (4) is converted into a digital signalq[m]By replacing with inverted digital signal sq_n[m]Then by sq_n[m]The frequency spectrum estimation expression of the left-increased double-edge two-level UPWM signal obtained by modulation is as follows:
Figure GDA0002955991880000051
and a sixth step: subtracting the formula (6) from the formula (5) and simplifying to obtain the digital modulation signal sq[m]And (3) modulating the obtained frequency spectrum estimation expression of the left-increased double-edge three-level UPWM signal:
Figure GDA0002955991880000052
compared with the prior art, the invention has the following advantages:
1. aiming at the condition that the input modulation signal is a digital signal, the frequency spectrum of a left-increased double-edge UPWM signal can be directly obtained according to the frequency spectrum estimation method provided by the invention;
2. according to the spectrum estimation expression provided by the invention, the distortion condition of the UPWM signal spectrum can be analyzed and researched from the aspect of mathematical theory, and theoretical guidance can be provided for constructing a high-performance digital D-type audio power amplifier;
3. the spectrum estimation method provided by the invention can directly obtain the spectrum amplitude of a certain frequency of the left-increased double-edge UPWM signal, and the whole spectrum is not required to be calculated aiming at the condition that the spectrum of a specific frequency is only required to be researched, so that the calculation and analysis processes are simpler and more convenient.
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FIG. 1 is a schematic structural diagram of a digital class D audio power amplifier;
FIG. 2 is a schematic diagram of the modulation principle and the dispersion and quantization process of left-augmented double-edged two-level and three-level UPWM signals (k)pwm＝8)；
FIG. 3 is a frequency spectrum of a left-growing double-edge two-level UPWM signal computed in accordance with the present invention;
FIG. 4 is a frequency spectrum of a left-growing double-edge three-level UPWM signal calculated in accordance with the present invention;
FIG. 5 is a measured frequency spectrum of a left-growing double-edge two-level UPWM signal;
FIG. 6 is a measured frequency spectrum of a left-growing double-edge three-level UPWM signal;
FIG. 7 is a graph of the THD versus digital PWM generator order k for a left-increasing double-edge two-level UPWM signal calculated in accordance with the present inventionpwmA graph of the results of the changes;
FIG. 8 is a graph of the THD versus digital PWM generator order k for a left-increasing double-edge three level UPWM signal calculated in accordance with the present inventionpwmGraph of the results of the changes.
Detailed Description
The process of constructing the frequency spectrum estimation expressions of the left-increasing double-edge two-level UPWM signal and the left-increasing double-edge three-level UPWM signal is fully described in detail below with reference to fig. 2 to 8.
A left-increasing double-edge UPWM signal frequency spectrum estimation method based on a digital PWM generator comprises the steps of firstly discretizing and quantizing a left-increasing double-edge two-stage UPWM signal output by the digital PWM generator on a time domain according to sampling frequency and quantization grade of the digital signal input by the digital PWM generator to obtain a series of discrete points; then, discrete Fourier transform is carried out on the discrete points by utilizing the distribution rule of the obtained discrete point amplitudes, and the transform result is multiplied by the frequency response function of the zero-order retainer to obtain a left-increased double-edge two-level UPWM signal frequency spectrum estimation expression; and subtracting the two-stage UPWM signal frequency spectrum estimation expressions obtained by modulating the input signal of the digital PWM generator before and after inversion to obtain the corresponding three-stage UPWM signal frequency spectrum estimation expressions.
The method comprises the following steps:
the first step is as follows: for reference analog signal s (t) at sampling frequency fsSampling and quantizing to obtain digital signal sq[m]T belongs to R, m belongs to Z, R represents real number, Z represents integer, k is used in quantization processqRepresenting the quantization level, kqEqual to the number of steps obtained by equally dividing the amplitude of s (t) into equal steps plus 1, and quantizing the digital signal sq[m]∈[0,kq-1]And s isq[m]∈Z；
The second step is that: the digital PWM generator converts the digital signal sq[m]Modulating into a left-increased double-edge two-level UPWM signal recorded as yGLDAD(t)；
The third step: will increase the left double-edged two-level UPWM signal yGLDADEach pulse period of (t) is equally divided in the time domain into kpwmPart, kpwmThe value of (A) usually satisfies the exponential form of 2, and kpwm＝kq-1，kpwmA number of stages, referred to as a digital PWM generator, also referred to as pulse width resolution; then for yGLDAD(t) discretization in the time domain, yGLDAD(t) each discrete point is distributed over the start time of each signal after division; then, carrying out quantization processing on the discrete points, wherein when the amplitude is high level, the value of the discrete point is 1, and when the amplitude is low level, the value of the discrete point is 0; corresponding to pair yGLDAD(t) at a sampling frequency fcSampling and quantizing, fc＝kpwm·fs(ii) a These discrete points can form a digital signal sequence, denoted as y, which approximates the original left-growing double-edge two-level UPWM signalGLDAD[n]N belongs to Z, N is more than or equal to 0 and less than or equal to N-1, and N is the total number of the discrete points; if a digital signal s is inputq[m]When the length of (A) is M, N is M.kpwm；
The fourth step: for the discrete point y obtained in the third stepGLDAD[n]The amplitude distribution rule is analyzed and researched, and the following results can be obtained:
Figure GDA0002955991880000071
i.e. when n ∈ [ mk ]pwm+kpwm/2-(sq[m]/2)up,mkpwm+kpwm/2+(sq[m]/2)down-1]When y isGLDAD[n]When n is other value, y is 1GLDAD[n]0; symbol ()upAnd ()downRepresents rounding up and down "+", respectively; for digital signal yGLDAD[n]Performing discrete Fourier transform to obtain digital signal yGLDAD[n]Fourier transform expression of (1), will yGLDAD[n]And the frequency response function h [ k ] of the zero-order keeper]Multiplying, k is more than or equal to 0 and less than or equal to N-1, k belongs to Z, and then obtaining the digital modulation signal sq[m]Frequency spectrum estimation expression y of modulated left-increased double-edge two-stage UPWM signalGLDAD[k](ii) a The frequency response function of the zeroth order keeper is generally expressed as:
Figure GDA0002955991880000072
wherein j is a unit of an imaginary number,
Figure GDA0002955991880000073
Tcis yGLDAD[n]Sampling period of (D), Tc＝1/fc＝1/(kpwm·fs)，Ω＝2πkfcN; simplifying h (j Ω) to obtain:
Figure GDA0002955991880000081
the spectral estimation expression of the left-growing double-edge two-level UPWM signal is:
Figure GDA0002955991880000082
formula (4) is simplified to obtain:
Figure GDA0002955991880000083
the fifth step: in order to obtain a frequency spectrum estimation expression of a left-increased double-edge three-level UPWM signal, an input digital signal s is firstly inputq[m]Taking the inverse to obtain a digital signal sq_n[m]The inverse process is sq_n[m]＝kpwm-sq[m]At this time, the digital signal s in the formula (4) is converted into a digital signalq[m]By replacing with inverted digital signal sq_n[m]Then by sq_n[m]The frequency spectrum estimation expression of the left-increased double-edge two-level UPWM signal obtained by modulation is as follows:
Figure GDA0002955991880000084
and a sixth step: subtracting the formula (6) from the formula (5) and simplifying to obtain the digital modulation signal sq[m]Frequency spectrum estimation expression y of modulated left-increased double-edge three-level UPWM signalGLDBD[k]The frequency spectrum estimation expression of the left-increased double-edge three-level UPWM signal is as follows:
Figure GDA0002955991880000091
the modulation principle and the discrete and quantization process schematic of the left-increased double-edge two-level UPWM signal and the left-increased double-edge three-level UPWM signal are shown in fig. 2. In FIG. 2, the reference analog signal s (t) is sampled at a frequency fsSampling at 384kHz and quantizing to obtain digital signal sq[m]Quantization level kqWhen the result is 9, sq[m]∈[0，8]. Digital PWM generatorq[m]Modulating into a left-increasing double-edge two-level UPWM signal yGLDAD(t) of (d). Discretizing and quantizing the left-increased double-edge two-stage UPWM signal, and firstly, performing discretization and quantization on each pulse period T of the left-increased double-edge two-stage UPWM signals(Ts＝1/fs2.6. mu.s) into kpwmPart, kpwm＝kq-1 ═ 8; then for yGLDAD(t) discretization in the time domain, yGLDAD(t) at each discrete point (y in FIG. 2)GLDADRed dots on (t) are distributed over the start time of each signal after division; then, carrying out quantization processing on the discrete points, wherein when the amplitude is high level, the value of the discrete point is 1, and when the amplitude is low level, the value of the discrete point is 0; corresponding to pair yGLDAD(t) at a sampling frequency fcSampling and quantizing, fc＝kpwm·fs＝3072kHz。yGLDAD(t) the discrete points obtained in the discretization process constitute a new digital signal yGLDAD[n]N belongs to Z, N is more than or equal to 0 and less than or equal to N-1, and N is the total number of the discrete points. If a digital signal s is inputq[m]When the length of (A) is M, N is M.kpwm8M. For digital signal yGLDAD[n]Making a discrete Fourier transform and converting the result of yGLDAD[n]The resulting discrete Fourier expression and the frequency response function h [ k ] of the zero order keeper]Multiplying, wherein k is more than or equal to 0 and less than or equal to N-1, k belongs to Z, and the obtained frequency spectrum estimation expression of the left-increased double-edge two-level UPWM signal is as follows:
Figure GDA0002955991880000092
wherein the frequency response function h [ k ] of the zero-order keeper is:
Figure GDA0002955991880000101
formula (8) is simplified to obtain:
Figure GDA0002955991880000102
to obtainFrom a digital signal sq[m]The obtained frequency spectrum estimation expression of the left-increased double-edge three-level UPWM signal needs to firstly obtain the digital signal sq[m]Negating to obtain a negated digital signal sq_n[m]，sq_n[m]＝kpwm-sq[m]＝8-sq[m]In this case, the digital signal s on the right side of the equal sign of the formula (8)q[m]By replacing with inverted digital signal sq_n[m]To obtain a digital signal sq_n[m]Modulating the obtained left-increased double-edge two-level UPWM signal spectrum expression:
Figure GDA0002955991880000103
subtracting the equation (11) from the equation (10) and simplifying the equationq[m]The obtained frequency spectrum estimation expression of the left-increased double-edge three-level UPWM signal is as follows:
Figure GDA0002955991880000104
when the input test signal has the amplitude of 0dBFS, the frequency of 6kHz and the sampling frequency fsTaking quantization level k in the case of sinusoidal signal of 384kHzq33, the number of stages k of the digital PWM generatorpwm＝kq-1 ═ 32, when the digital modulation signal s isq[m]And (4) determining. Interception s in simulation calculationq[m]Length M of 6400. According to the spectral estimation method proposed by the invention, the digital modulation signal s is calculated by MATLAB softwareq[m]The frequency spectrums of the modulated left-increased double-edge two-level UPWM signal and the modulated left-increased double-edge three-level UPWM signal are respectively shown in fig. 3 and fig. 4. A Field-Programmable Gate Array (FPGA) is utilized to build a left-increasing double-edge type digital PWM generator experiment test platform, and a digital modulation signal s is actually measuredq[m]The frequency spectrums of the modulated left-increased double-edge two-level UPWM signal and the modulated left-increased double-edge three-level UPWM signal are shown in fig. 5 and 6, respectively. Comparing FIG. 3 with FIG. 5 and FIG. 4 with FIG. 6, the root is knownAccording to the fact that the frequency spectrum obtained by the frequency spectrum estimation method is basically the same as the frequency spectrum result obtained by the FPGA experimental test platform, the frequency spectrum estimation method for the left-increased double-edge two-stage UPWM signal and the left-increased double-edge three-stage UPWM signal is reasonable and effective. Comparing fig. 3 and fig. 4, it can be seen that intermodulation distortion generated by the carrier and the input modulation signal in the left-increased double-edge three-level UPWM signal is greatly reduced near the odd harmonics of the carrier, so that the amplitude of the out-of-band high-frequency component is lower, and therefore the EMI (Electro-Magnetic Interference) generated by the left-increased double-edge three-level UPWM signal is lower.
When a modulated signal s is inputq[m]For sinusoidal digital signal with amplitude of 0dBFS and frequency of 1kHz (M is 6400), the Total Harmonic Distortion (THD) of left-increased double-edge two-stage UPWM signal and the THD of left-increased double-edge three-stage UPWM signal along with the number k of the digital PWM generator are given by using the frequency spectrum estimation method provided by the invention under the condition of different sampling frequencies of input modulation signalspwmThe results of the changes are shown in fig. 7 and 8, respectively. As can be seen from FIGS. 7 and 8, when the sampling frequency f of the modulated signal is setsWhen different values are taken, the THD of the left-increased double-edge two-level UPWM signal and the THD of the left-increased double-edge three-level UPWM signal are both along with kpwmIs increased and decreased, and when k is increasedpwmWhen smaller, the THD of both is substantially the same, when k ispwmWhen the value is larger than a certain value, the THD of the left-increased double-edge three-level UPWM signal is smaller than that of the left-increased double-edge two-level UPWM signal, which shows that the distortion of an output signal can be smaller and the fidelity is better by adopting a left-increased double-edge three-level UPWM modulation method in a digital D type audio power amplifier. Furthermore, as can be seen from a review of FIGS. 7 and 8, the THD of the left-increasing double-edge two-level UPWM signal and the THD of the left-increasing double-edge three-level UPWM signal are substantially a function of the sampling frequency fsIs increased and decreased, especially at kpwmIn the larger case, this tendency to decrease is more pronounced.
The spectrum estimation method can directly obtain the spectrum amplitude of a certain frequency of the left-increased double-edge UPWM signal, and the whole spectrum is not required to be calculated aiming at the condition that the spectrum of a specific frequency is only required to be researched, so that the calculation and analysis processes are simpler and more convenient.

Claims (2)

1. A left-increasing double-edge UPWM signal frequency spectrum estimation method based on a digital PWM generator is characterized in that: firstly, discretizing and quantizing a left-increased double-edge two-stage UPWM signal output by a digital PWM generator on a time domain according to the sampling frequency and the quantization grade of a digital signal input by the digital PWM generator to obtain a series of discrete points; then, discrete Fourier transform is carried out on the discrete points by utilizing the distribution rule of the obtained discrete point amplitudes, and the transform result is multiplied by the frequency response function of the zero-order retainer to obtain a left-increased double-edge two-level UPWM signal frequency spectrum estimation expression; and subtracting the two-stage UPWM signal frequency spectrum estimation expressions obtained by modulating the input signal of the digital PWM generator before and after inversion to obtain the corresponding three-stage UPWM signal frequency spectrum estimation expressions.
2. The method of estimating a left-growing double-edge UPWM signal spectrum based on a digital PWM generator according to claim 1, wherein: the method comprises the following steps:
the first step is as follows: for reference analog signal s (t) at sampling frequency fsSampling and quantizing to obtain digital signal sq[m]T belongs to R, m belongs to Z, R represents real number, Z represents integer, k is used in quantization processqRepresenting the quantization level, kqEqual to the number of steps obtained by equally dividing the amplitude of s (t) into equal steps plus 1, and quantizing the digital signal sq[m]∈[0,kq-1]And s isq[m]∈Z；
The second step is that: the digital PWM generator converts the digital signal sq[m]Modulating into a left-increased double-edge two-level UPWM signal recorded as yGLDAD(t)；
The third step: will increase the left double-edged two-level UPWM signal yGLDADEach pulse period of (t) is equally divided in the time domain into kpwmPart, kpwmThe value of (A) usually satisfies the exponential form of 2, and kpwm＝kq-1，kpwmNumber of stages, also called digital PWM generatorsPulse width resolution; then for yGLDAD(t) discretization in the time domain, yGLDAD(t) each discrete point is distributed over the start time of each signal after division; then, carrying out quantization processing on the discrete points, wherein when the amplitude is high level, the value of the discrete point is 1, and when the amplitude is low level, the value of the discrete point is 0; corresponding to pair yGLDAD(t) at a sampling frequency fcSampling and quantizing, fc＝kpwm·fs(ii) a These discrete points can form a digital signal sequence, denoted as y, which approximates the original left-growing double-edge two-level UPWM signalGLDAD[n]N belongs to Z, N is more than or equal to 0 and less than or equal to N-1, and N is the total number of the discrete points; if a digital signal s is inputq[m]When the length of (A) is M, N is M.kpwm；
The fourth step: for yGLDAD[n]The distribution rule of discrete point amplitude is analyzed and researched, and the following results can be obtained:
Figure FDA0002883584330000021
wherein the symbol (#)upAnd ()downRepresents rounding up and down "+", respectively; for digital signal yGLDAD[n]Performing discrete Fourier transform, and then combining the obtained Fourier transform expression with the frequency response function h [ k ] of the zero-order keeper]Multiplying, k is more than or equal to 0 and less than or equal to N-1, k belongs to Z, and obtaining a digital modulation signal sq[m]And the frequency spectrum estimation expression of the modulated left-increased double-edge two-stage UPWM signal is recorded as yGLDAD[k](ii) a The frequency response function of the zeroth order keeper is generally expressed as:
Figure FDA0002883584330000022
wherein j is a unit of an imaginary number,
Figure FDA0002883584330000023
Tcis yGLDAD[n]Sampling period of (D), Tc＝1/fc＝1/(kpwm·fs)，Ω＝2πkfcN; simplifying h (j Ω) to obtain:
Figure FDA0002883584330000024
the spectral estimation expression of the left-growing double-edge two-level UPWM signal is:
Figure FDA0002883584330000025
formula (4) is simplified to obtain:
Figure FDA0002883584330000031
the fifth step: in order to obtain a frequency spectrum estimation expression of a left-increased double-edge three-level UPWM signal, an input digital signal s is firstly inputq[m]Taking the inverse to obtain a digital signal sq_n[m]The inverse process is sq_n[m]＝kpwm-sq[m]At this time, the digital signal s in the formula (4) is converted into a digital signalq[m]By replacing with inverted digital signal sq_n[m]Then by sq_n[m]The frequency spectrum estimation expression of the left-increased double-edge two-level UPWM signal obtained by modulation is as follows:
Figure FDA0002883584330000032
and a sixth step: subtracting the formula (6) from the formula (5) and simplifying to obtain the digital modulation signal sq[m]And (3) modulating the obtained frequency spectrum estimation expression of the left-increased double-edge three-level UPWM signal:
Figure FDA0002883584330000033
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