## 1 Introduction  
In recent years, by virtue of their high efficiency, high reliability and superior speed regulation performance, permanent magnet synchronous motors (PMSMs) have been widely used in the development of new energy vehicles [1; 2]. The drive system of a PMSM typically uses the voltage source inverter (VSI) as the hardware-based control system [3]. However, the VSI drive makes the motor line voltage and phase current rich in harmonic components [4; 5]. Usually, conventional pulse-width modulation (PWM) techniques employ fixed carrier frequency modulation, resulting in the sideband harmonic current spectrum concentrated in the carrier frequency and its multiples [6; 7; 8], thereby causing a high-frequency, uncomfortable \"whistling\" radiated from the motor [9], presenting a challenge to the vehicle's noise, vibration and harshness (NVH) performance [10]. In order to effectively suppress the harmonic components of sidebands, many scholars, both domestic and abroad, have conducted research on optimizing the harmonic current of the inverter. [PERSON] et al. [11] analyzed the current's harmonic characteristics introduced by PWM and derived the corresponding analytical model of radial electromagnetic force. [PERSON] et al. [12] investigated the basic characteristics of on-line and off-line PWM techniques and their effects on the electromagnetic noise of motors, validating the accuracy of the theoretical analysis through experiments under variable speed conditions. [PERSON] et al. [13] proposed a method for implementing variable-delay random pulse-width modulation in automotive traction drive inverters, which can reduce electromagnetic radiation from the inverter and minimize electromagnetic compatibility filtering to decrease the noise generated by the drive system. [PERSON] et al. [14] proposed a new method of combining a dual three-phase PMSM with a carrier phase-shift mechanism to substantially reduce the odd-order PWM frequency vibration. [PERSON] et al. [15] analyzed the effects of spread spectrum modulation based on sinusoidal waves. Although the periodic spread spectrum modulation technique allows the originally fixed carrier frequency to be varied in a deterministic and controllable manner, the sideband harmonic suppression is limited. [PERSON] et al. [16] analyzed the relationship between sideband current harmonics and mechanical vibration noise in different carrier frequency domains, which provides a theoretical foundation for the prediction of the vibro-acoustics of PMSMs. However, the studies on current harmonics and the vibro-acoustic response are mostly based on continuous pulse-width modulation (CPWM) [17; 18], and the characteristic laws of sideband harmonics generated by the DPWM strategies are not yet well studied. The DPWM can significantly improve the inverter efficiency and has been widely applied in high-power voltage source inverters [19; 20], but it produces richer sideband harmonic components. Despite its effectiveness in suppressing sideband harmonic components, random carrier frequency modulation (RCFM) has a flaw when it comes to total harmonic distortion [21]. Conventional RCFM techniques usually employ pseudo-random number generation methods based on the use of linear congruence or look-up table methods [22]. Compared with ideal random numbers, pseudo-random numbers may be larger than or smaller than the mathematical expectation within local time periods, which results in the generated randomized carrier frequency being successively larger or smaller than the initial carrier frequency [23]. This imbalance will not maximize the suppression of sideband harmonics to further suppress the sideband vibro-acoustics [24; 25; 26], and it will result in large current ripples in the output signal of the system, thereby causing output torque fluctuations and the generation of additional switching losses [27]. To this end, this paper proposes a Markov chain random carrier frequency modulation (MRCFM) technique to promote the inhibitory effect of sideband harmonic current and vibration noise.