A new method for DDS spurious suppression based on second-order phase perturbation

The basic principle and spurious sources of DDS are introduced , the causes of phase truncation spurious and the principle of common phase perturbation are analyzed, and on this basis, an improved second-order phase perturbation method is proposed. The method is derived and demonstrated in this paper. The study found that the suppression effect of DDS spurious using this method is more significant than that of the common phase perturbation method, reaching 18 dB per phase position. Finally, the feasibility of the method is verified by simulation using DSP Builder in Matlab.

1 Basic Principles of DDS and Spurious Analysis

1.1 Basic Principles of DDS

DDS directly searches the storage table to obtain the amplitude value of the output waveform corresponding to each phase, and changes the output frequency by changing the sampling frequency and phase step. Its principle structure is shown in Figure 1.

In Figure 1, the phase accumulator accumulates with a step size of K under the control of the clock frequency fc, outputs an N-bit quantized phase sequence, and then takes its high W bits as the addressing address of the ROM to address the lookup table ROM. The L-bit discrete amplitude sequence output by the addressing is converted into a step wave by the DAC, and then smoothed by a low-pass filter (LPF) to obtain a synthesized signal waveform. Output frequency

.

1.2 Spurious analysis

The working principle of DDS determines that its output is rich in spurious signals, of which the main sources are three aspects: (1) Phase truncation error εp(n), using the high W bits of the N-bit phase accumulator for addressing, truncating the low B=NW bits. This introduces a phase truncation error. (2) Amplitude quantization error εA(n), the sine value stored in the ROM is represented by a limited L bits, which produces an amplitude quantization error. (3) DAC conversion error εDA(n), caused by the non-ideal characteristics of the actual DAC device.

Among the three sources of DDS spurious, phase truncation and DAC conversion have the greatest impact, but the spurious model caused by DAC conversion cannot be established at present, so under the assumption that the other two spurious sources do not exist, the spurious introduced by phase truncation is mainly studied. When there is no phase truncation, the ROM table output sequence S(n) can be obtained.

It can be seen that in addition to the desired signal, the digital spectrum output by the DDS also contains spurious components after εp(n) is modulated by the cosine signal. According to the derivation in the literature, the output signal-to-noise ratio of the DDS under phase truncation satisfies

From formula (4), it can be seen that when the number of truncation bits B is reduced by 1, the spurious signal is improved by about 6dB.

2 Phase perturbation method to suppress spurious signals

2.1 Principle of general phase perturbation

The common phase perturbation technology is to add a random signal that satisfies certain statistical characteristics to the output of the phase accumulator after each clock pulse arrives to break the periodicity of the error sequence, thereby reducing spurious signals. The principle is shown in Figure 2.

In Figure 2, the N-bit phase sequence φ(n) is added to the B-bit disturbance sequence Z(n), and then truncated to W bits through phase. The phase truncation process can be regarded as a quantization process with a quantization interval of △=2-w. The output signal after truncation is φ(n)+Z(n)+ep(n), ep(n) is the phase quantization error, and the total quantization noise ε(n)=Z(n)+ep(n) is the sum of the disturbance signal and the phase quantization error. According to the perturbation quantization method in the literature, Z(n) and ep(n) are both white noises that obey a uniform distribution in [-△/2, △/2]. Their sum ε(n) is uncorrelated with φ(n) and is white. Therefore, this added disturbance sequence can make the quantization error independent of the original input signal and become white noise that obeys a uniform distribution.

From the above analysis, we know that the DDS output signal with the disturbance signal added is X(n)=sin(2π(φ(n)+ε(n))). Assume that the above formula is expanded by Taylor series at 2πfn:

By using the ordinary phase perturbation method, the suppression of spurious components can be increased from 6 dB per phase position to 12 dB.

2.2 Improved second-order phase perturbation method

The second-order phase perturbation method is formed on the basis of studying the ordinary phase perturbation method. In this method, the perturbation sequence is generated by adding two independent and identically distributed random sequences. The specific principle structure is shown in Figure 3.

As shown in FIG3 , two B-bit independent and identically distributed random sequences are added to generate a B+1-bit disturbance sequence, and then the B+1-bit disturbance sequence is used to perturb the original output φ(n), which can achieve a better spurious suppression effect.

For the second-order phase disturbance, it is necessary to consider the third-order moment component E{ε3(n)} of the quantization noise. The Taylor series expansion of the output signal is:

Assuming that the perturbation sequence is the sum of two random sequences that obey a uniform distribution on [0, △], the probability density of the perturbation sequence is

The disturbance sequence satisfying equation (14) is added to the phase sequence and truncated to W bits. The resulting total quantization noise has three cases:

From formula (17), it can be seen that by using the second-order phase perturbation method, the suppression of spurious components can reach 18 dB per phase position, which is a significant improvement compared to the ordinary phase perturbation method.

3 Simulation Verification

The simulation is performed using the DSP Builder tool embedded in Matlab, and the specific model is shown in Figure 4. Simulation parameters: clock frequency fc = 1 MHz; frequency control word K = 485 952; phase accumulator bit number N = 22; phase addressing bit number W = 4; ROM output bit number L = 20; the sum of two independent 24-level 18-bit output m sequences is taken as the disturbance sequence. The simulation results are sent to the Matlab workspace and power spectrum transformation is performed to verify the design of the system.

Figure 5 is a graph obtained by normalizing the power spectrum of the DDS system simulation results in three cases. Figure 5(a) shows the system output power spectrum without any phase disturbance. Figure 5(b) shows the system output power spectrum after adding a 24-level 18-bit output m-sequence as the disturbance sequence. Figure 5(c) shows the system output power spectrum in this case when the sum of two 24-level 18-bit output m-sequences is taken as the disturbance sequence. It can be seen from the figure that since the number of phase addressing bits is 4, the maximum spurious signal is -24.2 dBc without phase disturbance, -46.8 dBc with ordinary phase disturbance, and -67.7 dBc with second-order phase disturbance, which is different from the theoretically derived -72 dBc. This is due to the influence of the point number limit when performing FFT. From the above data, it can be concluded that the spurious suppression performance of DDS is greatly improved by using the second-order phase disturbance method.

4 Conclusion

Based on the study of the basic phase perturbation method, a new second-order phase perturbation method is proposed, which can suppress the spurious component to 18 dB per phase position. Therefore, under the same spurious accuracy requirement, the design using this method can reduce the number of ROM addressing bits, compress the ROM storage space, and reduce the hardware design complexity and product cost.