High-precision DDS quantization error suppression method based on curvature adaptive segmentation

Abstract

The invention provides a high-precision DDS quantization error suppression method based on curvature adaptive segmentation. According to the technical scheme, the method comprises the steps that firstly, the curvature of a sine function at different positions and the bending degree of a function curve are analyzed, and optimal segmentation processing is conducted on the function; 2, selecting interpolation selection of a high-curvature region by analyzing a linear interpolation error and a quadratic interpolation error; and 3, selecting a segmentation density and interpolation method by setting a target error. And 4, performing parameter optimization on the secondary interpolation of the high curvature region through a Minimax criterion and a Levenberg-Marquardt iterative algorithm. And 5, calculating a frequency control word of the high curvature area by using the iteratively optimized data. And 6, calculating a conventional sine frequency control word of a low curvature area and generating a signal through a lookup table ROM. According to the invention, the quantization error of the traditional DDS is reduced, the maximum amplitude error is reduced to 0.1 LSB from 0.5 LSB (12 bits) of the traditional linear interpolation, the curvature is adaptively segmented, the Minimax quadratic interpolation is optimized, and the problem of high curvature region error is solved.

Description

Technical Field

[0001] The present invention belongs to the technical field of digital signal synthesis and high-precision radio frequency systems, and in particular relates to a high-precision DDS quantization error suppression method based on curvature adaptive segmentation. Background Art

[0002] Direct digital frequency synthesizer (DDS) is a key digital technology. Compared to PLLs, DDS offers superior frequency resolution, fast, phase-continuous frequency conversion, and excellent stability. However, due to its fully digital architecture, DDS inevitably introduces spurious signals. These spurious signals originate primarily from three sources: spurious signals caused by phase truncation errors in the phase accumulator; spurious signals caused by amplitude quantization errors (caused by the limited word length of the memory); and spurious signals caused by non-ideal DAC characteristics. In low-frequency DDS circuits, amplitude quantization errors are a key factor affecting DDS spectrum quality. The following article discusses amplitude quantization errors in low-frequency DDS circuits and their mitigation methods. Since sine waves are continuous, smooth analog waveforms, they require an infinite number of binary values ​​to accurately represent them. However, the storage capacity of lookup tables (ROMs) is limited, and the resolution of DACs also limits the number of quantization bits that can be used to represent the signal waveform. Therefore, in DDS systems, the number of bits (W) representing the binary value of a sine wave amplitude is generally relatively small and finite, leading to amplitude quantization errors. Summary of the Invention

[0003] In view of the shortcomings of the prior art described above, the object of the present invention is to provide a high-precision DDS quantization error suppression method based on curvature adaptive segmentation, which is used to solve the problem in the prior art that the amplitude quantization error of the direct digital frequency synthesizer (DDS) caused by the limited word length of the memory lookup table ROM and the linear interpolation error of the traditional DDS increases with the square of the step length, making it unable to meet the high-precision requirements of the peak / valley.

[0004] To achieve the above objectives and other related objectives, the present invention provides a high-precision DDS quantization error suppression method based on curvature adaptive segmentation.

[0005] A high-precision DDS quantization error suppression method based on curvature adaptive scoring comprises the following steps:

[0006] Step 1: By analyzing the curvature of the sine function at different positions and the degree of curvature of the function curve, the function is optimally segmented;

[0007] Step 2: Select the interpolation option for the high curvature area by analyzing the linear interpolation error and the quadratic interpolation error;

[0008] Step 3: Select the segment density and interpolation method by setting the target error, according to the target maximum error Δy max Backward derivation step size h;

[0009] Step 4: Optimize the quadratic interpolation of high curvature areas based on the Minimax criterion and the Levenberg-Marquardt algorithm;

[0010] Step 5: Output FTW according to the final iteration result;

[0011] Step 6: Calculate the regular sinusoidal frequency control word in the low curvature region and generate the signal through the lookup table ROM.

[0012] Preferably, the high and low curvature regions are divided by analyzing the curvature segmentation of the sine function. The second derivative of the sine function y=sin(θ) is y″=-sin(θ), and its curvature formula is simplified to:

[0013]

[0014] Peak / trough region (θ = 90°, 270°, etc.): sin(θ) = ±1, curvature reaches its maximum value k max = 1, the function is most curved. Flat regions (θ = 0°, 180°, 360°, etc.): sin(θ) = 0, k = 0, and the function approximates a line segment. High-curvature regions: The function is sharply curved. If the number of segments is insufficient, linear or low-order polynomial approximations will introduce significant errors. Low-curvature regions: The function approximates a straight line. Sparse segmentation can meet accuracy requirements and save resources. Quantitative analysis of segment density and maximum error is also performed.

[0015] Preferably, the interpolation method for high curvature areas is selected by analyzing and comparing the errors of linear interpolation and quadratic interpolation. For the function f(x) on the interval [a, b], use the nth degree polynomial P n (x) is interpolated, and the error (remainder) is:

[0016]

[0017] where x k is the interpolation node. In the interval [a,b], the maximum error satisfies:

[0018]

[0019] Where m is the number of segments, is the step length.

[0020] Single-segment linear approximation error: Assume that in the interval [θ i ,θ i+1 ]Use the straight line y=kθ+b to approximate y=sin(θ) with a step size of h=θ i+1 -θ i The point where the derivative is zero within the interval where the maximum error occurs satisfies:

[0021]

[0022] Through Taylor expansion analysis, the maximum error can be approximated as:

[0023]

[0024] If the sine period 2π is divided into m segments, the step size is The maximum error is: If the error optimization of curvature adaptive segmentation is used, then in the high curvature area (for example, θ≈π / 2), the curvature k≈1 and the step size h high Small; in low curvature regions (e.g., θ≈0), k≈0, and the step size h low Larger. Total number of segments m = m high +m low ,in:

[0025]

[0026] where K high is the proportion of the interval with high curvature, K low is the proportion of the interval with low curvature.

[0027] The largest errors are dominated by regions of high curvature: Error analysis of quadratic polynomial interpolation of sine function:

[0028] (1) Single-segment quadratic approximation error

[0029] Using the quadratic polynomial y = aθ 2 +bθ+c approximates the sine function, and its maximum error remainder is:

[0030]

[0031] because The upper limit of error is:

[0032] (2) Relationship between error and segment density

[0033] Uniform segmentation: the total error is determined by the maximum step size h max Decide:

[0034]

[0035] Curvature adaptive segmentation: step size h in high curvature area high , low curvature step length h low , the total error is:

[0036]

[0037] Linear interpolation: The error is proportional to the square of the step size, Δy∝h 2 . Quadratic interpolation: The error is proportional to the cube of the step size, Δy∝h 3 Curvature adaptive segmentation dynamically adjusts h. Under the same number of segments, h in the high curvature area is reduced to 1 / k of the original value, and the error is reduced to:

[0038]

[0039] For example, if the step size is halved (k=2) in the high curvature region, the linear interpolation error is reduced to 1 / 4 and the quadratic interpolation error is reduced to 1 / 8. linear ≈0.5LSB, the error after quadratic interpolation optimization is Δy qyad ≈0.1LSB, the power error is reduced by a factor of Corresponding SFDR gain: 10·log 10 (25)≈14dB.

[0040] Preferably, the polynomial coefficients are optimized by quadratic interpolation in high curvature areas, and the optimal parameters are obtained by iterative Levenberg-Marquardt algorithm, which includes discretized sampling and uniformly selecting N sampling points in the interval. Calculate the corresponding ideal amplitude y k = sinθ k ; Define the slack variable t and convert the Minimax problem into a constrained optimization problem:

[0041]

[0042] By introducing the penalty function, the constrained problem is transformed into an unconstrained optimization problem, and the residual vector r is defined k :

[0043]

[0044] The objective function is:

[0045]

[0046] Use the Levenberg-Marquardt algorithm iteration: (1) Use the Taylor expansion or linear interpolation result as the initial value, the initial slack variable: set (2) Residual vector r = [r1, r2…, r N ] T For parameters p = [a, b, c, t] T The Jacobian matrix J is:

[0047]

[0048] where e k = sinθ k -(aθ+bθ+c);

[0049] (3) Iteratively update parameter p: p (n+1) =p (n) -(J T J+λdiag(J T J)) -1 J T (4) When the residual change rate |Δr| / |r| < ε, the parameter change rate |Δp| / |p| < ε, or the maximum number of iterations is reached, the iteration is stopped to obtain the optimized parameter results.

[0050] As described above, the high-precision DDS quantization error suppression method based on curvature adaptive segmentation of the present invention has the following beneficial effects:

[0051] 1. The influence of different interpolation methods and segmentation density on the error in the high curvature region of the sine function is theoretically analyzed. The upper Minimax criterion and the Levenberg-Marquardt iterative algorithm are proposed to optimize the parameters of the quadratic interpolation to overcome the difficulty of error calculation in the high curvature region.

[0052] 2. The present invention effectively reduces the quantization error of the signal waveform caused by the limited storage capacity of the lookup table ROM and the limited resolution of the DAC, thereby improving the signal quality.

[0053] 3. The present invention reduces the maximum amplitude error from 0.5LSB (12 bits) of traditional linear interpolation to 0.1LSB through curvature adaptive segmentation, improves SFDR from the traditional 72dBc to 86dBc, and reduces spurious energy by 14dB. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 Flowchart of curvature-adaptive segmented DDS signal generation.

[0055] Figure 2 Flowchart for optimization of quadratic polynomial coefficients.

[0056] Figure 3 Levenberg-Marquardt algorithm iterative flowchart.

[0057] Figure 4 Dual DDS generation schematic.

[0058] In order to more clearly illustrate the embodiments of the present invention or the technical solutions of the prior art, the following is a brief introduction to the drawings required for use in the prior art. It is obvious that the drawings described below are only some embodiments of the present invention. DETAILED DESCRIPTION

[0059] The following describes the embodiments of the present invention through specific examples. Those skilled in the art will readily understand the other advantages and benefits of the present invention from the disclosure herein. The present invention may also be implemented or applied through various other specific embodiments, and the details in this specification may be modified or altered based on different viewpoints and applications without departing from the spirit of the present invention.

[0060] Please refer to the accompanying drawings. It should be noted that the illustrations provided in this embodiment are merely schematic illustrations of the basic concept of the present invention. Therefore, the drawings only show components relevant to the present invention and are not drawn according to the number, shape, and size of components in actual implementation. In actual implementation, the type, quantity, and proportion of each component may be varied arbitrarily, and the component layout may also be more complex.

[0061] This invention provides a high-precision DDS quantization error suppression method based on curvature adaptive segmentation. It aims to address the amplitude quantization error caused by the limited word length of the memory lookup table (ROM) in traditional direct digital synthesizers (DDSs). This method also addresses the linear interpolation error of traditional DDSs, which increases with the square of the step size and cannot meet the high-precision requirements of peaks and valleys. Specifically, the method analyzes the curvature of a function curve to divide it into high and low curvature regions, selects the interpolation method and segmentation density, and uses an algorithm to optimize parameters to resolve the quantization error and improve signal quality.

[0062] Specific examples

[0063] Step 1: By analyzing the curvature of the sine function at different positions and the degree of curvature of the function curve, the function is optimally segmented. The specific analysis is as follows:

[0064] Curvature distribution of the sine function:

[0065] The second derivative of the sine function y = sin(θ) is y″ = -sin(θ), and its curvature formula is simplified to:

[0066]

[0067] Peak / trough region (θ = 90°, 270°, etc.): sin(θ) = ±1, curvature reaches its maximum value k max = 1, the function is most curved. Flat area (θ = 0°, 180°, 360°, etc.): sin(θ) = 0, k = 0, the function changes approximately like a line segment.

[0068] In high-curvature regions, the function curves sharply. If the number of segments is insufficient, linear or low-order polynomial approximations can introduce significant errors. In low-curvature regions, the function approximates a straight line, so sparse segmentation can meet accuracy requirements and save resources. Quantitative analysis of segment density and maximum error is also performed.

[0069] For the function f(x) on the interval [a,b], use the nth degree polynomial P n (x) is interpolated, and the error (remainder) is:

[0070]

[0071] where x k is the interpolation node. In the interval [a,b], the maximum error satisfies:

[0072]

[0073] Where m is the number of segments, is the step length.

[0074] Step 2: Select the interpolation method for high curvature areas by analyzing the linear interpolation error and the quadratic interpolation error.

[0075] The linear interpolation error analysis of the sine function is as follows:

[0076] Single-segment linear approximation error: Assume that in the interval [θ i ,θ i+1 ]Use the straight line y=kθ+b to approximate y=sin(θ) with a step size of h=θ i+1 -θ i The point where the derivative is zero within the interval where the maximum error occurs satisfies:

[0077]

[0078] Through Taylor expansion analysis, the maximum error can be approximated as:

[0079] If the sine period 2π is divided into m segments, the step size is The maximum error is:

[0080] If the error optimization of curvature adaptive segmentation is used, then in the high curvature area (for example, θ≈π / 2), the curvature k≈1 and the step size h high Small; in low curvature regions (e.g., θ≈0), k≈0, and the step size h low Larger. Total number of segments m = m high +m low ,in:

[0081]

[0082] where K high is the proportion of the interval with high curvature, K low is the proportion of the interval with low curvature.

[0083] The largest errors are dominated by regions of high curvature: Error analysis of quadratic polynomial interpolation of sine function:

[0084] (1) Single-segment quadratic approximation error

[0085] Using the quadratic polynomial y = aθ 2 +bθ+c approximates the sine function, and its maximum error remainder is:

[0086]

[0087] because The upper limit of error is:

[0088] (2) Relationship between error and segment density

[0089] Uniform segmentation: the total error is determined by the maximum step size h max Decide:

[0090]

[0091] Curvature adaptive segmentation: step size h in high curvature area high , low curvature step length h low , the total error is:

[0092] Summarize:

[0093] Linear interpolation: The error is proportional to the square of the step size, Δy∝h 2 . Quadratic interpolation: The error is proportional to the cube of the step size, Δy∝h 3 Curvature adaptive segmentation dynamically adjusts h. Under the same number of segments, h in the high curvature area is reduced to 1 / k of the original value, and the error is reduced to:

[0094] For example, if the step size is halved (k=2) in the high curvature region, the linear interpolation error is reduced to 1 / 4 and the quadratic interpolation error is reduced to 1 / 8. The maximum linear interpolation error at 12-bit quantization is Δy linear ≈0.5LSB, the error after quadratic interpolation optimization is Δy qyad ≈0.1LSB, the power error is reduced by a factor of Corresponding SFDR gain: 10·log 10 (25)≈14dB.

[0095] Step 3: Select the segment density and interpolation method by setting the target error. max Backward deduction of step length h. The analysis is as follows:

[0096] Linear interpolation:

[0097] Quadratic interpolation:

[0098] Example (target error Δy max =0.1LSB):

[0099] Linear interpolation

[0100] Quadratic interpolation Conclusion: Quadratic interpolation is used in high curvature areas to meet the error requirements with fewer segments.

[0101] Step 4: Optimize the quadratic interpolation of high curvature areas based on the Minimax criterion and the Levenberg-Marquardt algorithm. The specific process is as follows:

[0102] Step 4.1 Polynomial coefficient optimization

[0103] In the interval [θ i ,θ i+1 ], find the quadratic polynomial P2(θ)=aθ 2 +bθ+c, so that the maximum absolute error is minimized:

[0104]

[0105] Step 4.1.1: Discrete sampling uniformly selects N sampling points within the interval (usually N ≥ 100), calculate the corresponding ideal amplitude y k = sinθ k .

[0106] Step 4.1.2: Introducing slack variables

[0107] Define the slack variable t and convert the Minimax problem into a constrained optimization problem:

[0108]

[0109] Step 4.1.3: Convert to nonlinear least squares form

[0110] By introducing the penalty function, the constrained problem is transformed into an unconstrained optimization problem. Define the residual vector r k :

[0111]

[0112] The objective function is:

[0113]

[0114] The flowchart of quadratic polynomial coefficient optimization is as follows Figure 2 shown.

[0115] Step 4.2 Levenberg-Marquardt algorithm iteration process

[0116] Step 4.2.1: Parameter initialization

[0117] Initial guess: Use Taylor expansion or linear interpolation as the initial value. For example, near the peak (θ = π / 2):

[0118]

[0119] Initial slack variables: Let

[0120] Step 4.2.2: Jacobian matrix calculation

[0121] Residual vector r=[r1,r2…,r N ] T For parameters p = [a, b, c, t] T The Jacobian matrix J is:

[0122]

[0123] where e k = sinθ k -(aθ+bθ+c)

[0124] Step 4.2.3: Parameter Update

[0125] Iteratively update parameter p: p (n+1) =p (n) -(J T J+λdiag(J T J)) -1 J T r

[0126] Damping factor adjustment: If the residual decreases, accept the update and decrease λ (e.g., λ: = λ / 10). If the residual increases, reject the update and increase λ (e.g., λ: = λ×10).

[0127] Step 4.2.4: Convergence judgment

[0128] The iteration is terminated when any of the following conditions are met:

[0129] (1) Residual change rate |Δr| / |r|<ε(e.g. ε=10 -6 ).

[0130] (2) Parameter change rate |Δp| / |p|<ε.

[0131] (3) Reach the maximum number of iterations (e.g., 100).

[0132] The iterative flow chart of the Levenberg-Marquardt algorithm is as follows Figure 3 shown.

[0133] Step 4.3 Specific optimization example target interval: θ∈[π / 2-0.2,π / 2+0.2] (approximately ±11.5°)

[0134] Sampling points: N = 200

[0135] Initial parameter: a (0) =-0.5,b (0) =0,c (0) =1,t (0) =0.02 Iteration process:

[0136] 1 First iteration:

[0137] Calculate the residual r (0) =0.018

[0138] Jacobian matrix J dimension: 200×4

[0139] Updated parameters: a (1) =-0.498,b (1) =0.003,c (1) =0.999,t (1) =0.015

[0140] The residual decreases and receives an update, λ=0.1

[0141] 2 Fifth iteration:

[0142] Residual r (5) =0.0005

[0143] Parameter change rate <10 -4 , terminate the iteration.

[0144] Optimization results:

[0145] a=-0.497, b=0.002, c=0.999, t=0.0005

[0146] The maximum error is 0.0005 (corresponding to a 12-bit quantization error of 0.002LSB)

[0147] Step 5: Calculate and output the FTW of the high curvature area based on the final iteration result.

[0148] Step 6: Output the low curvature area FTW and generate the corresponding signal waveform through the lookup table. The DDS generation principle diagram is as follows Figure 4 shown.

[0149] In summary, the present invention's high-precision DDS quantization error suppression method based on curvature adaptive segmentation has the following beneficial effects: It reduces the quantization error of traditional DDS. Through curvature adaptive segmentation, the maximum amplitude error is reduced from 0.5LSB (12-bit) using traditional linear interpolation to 0.1LSB, the SFDR is improved from the traditional 72dBc to 86dBc, and spurious energy is reduced by 14dB. The curvature adaptive segmentation and Minimax quadratic interpolation optimization overcome the error problem in high-curvature regions.

[0150] Therefore, the present invention effectively overcomes various shortcomings of the prior art and has high industrial utilization value.

[0151] The above embodiments are merely illustrative of the principles and effects of the present invention and are not intended to limit the present invention. Anyone skilled in the art may modify or alter the above embodiments without departing from the spirit and scope of the present invention. Therefore, all equivalent modifications or alterations made by one of ordinary skill in the art without departing from the spirit and technical principles disclosed herein are intended to be covered by the claims of the present invention.

Claims

1. A high-precision DDS quantization error suppression method based on curvature adaptive segmentation, characterized in that: The following steps are involved: Step 1: By analyzing the curvature of the sine function at different positions and the degree of curvature of the function curve, the function is optimally segmented; Step 2: Select the interpolation option for the high curvature area by analyzing the linear interpolation error and the quadratic interpolation error; Step 3: Select the segment density and interpolation method by setting the target error, according to the target maximum error Δy max Backward derivation step size h; Step 4: Optimize the quadratic interpolation of high curvature areas based on the Minimax criterion and the Levenberg-Marquardt algorithm; Step 5: Output FTW according to the final iteration result; Step 6: Calculate the regular sinusoidal frequency control word in the low curvature region and generate the signal through the lookup table ROM.

2. The high-precision DDS quantization error suppression method based on curvature adaptive segmentation according to claim 1, characterized in that: The function curve can be divided into high and low curvature areas according to different curvatures, and multiple segments can be used in the high curvature area, and sparse segments can be used to approximate straight lines in the low curvature area to save resources.

3. The high-precision DDS quantization error suppression method based on curvature adaptive segmentation according to claim 1, characterized in that: The maximum error of linear interpolation in high curvature areas is The maximum error of quadratic interpolation is Linear interpolation: The error is proportional to the square of the step size, Δy∝h 2 ; Quadratic interpolation: the error is proportional to the cube of the step size, Δy∝h 3 ; Curvature adaptive segmentation dynamically adjusts h. Under the same number of segments, h in the high curvature area is reduced to 1 / k of the original value, and the error is reduced to:

4. The high-precision DDS quantization error suppression method based on curvature adaptive segmentation according to claim 1, characterized in that: Uniform segmentation error Error of curvature adaptive segmentation Inverse step size by target maximum error: linear interpolation: Quadratic interpolation: It can be seen that using quadratic interpolation in high curvature areas can meet the error requirements with fewer segments.

5. The high-precision DDS quantization error suppression method based on curvature adaptive segmentation according to claim 1, characterized in that: The quadratic interpolation optimization includes the following specific steps: Step 4.1: Polynomial coefficient optimization, in the interval [θ i ,θ i+1 ], find the quadratic polynomial P2(θ)=aθ 2 +bθ+c, so that the maximum absolute error is minimized; Step 4.2: Use the Levenberg-Marquardt algorithm to iteratively obtain the optimal parameters.

6. The high-precision DDS quantization error suppression method based on curvature adaptive segmentation according to claim 5, characterized in that: The quadratic polynomial optimization comprises the following specific steps: Step 6.1: Discrete sampling, uniformly select N sampling points in the interval Calculate the corresponding ideal amplitude y k = sinθ k ; Step 6.2: Define the slack variable t and convert the Minimax problem into a constrained optimization problem: Step 6.3: By introducing a penalty function, the constrained problem is transformed into an unconstrained optimization problem and the residual vector r is defined. k : The objective function is:

7. The high-precision DDS quantization error suppression method based on curvature adaptive segmentation according to claim 5, characterized in that: The Levenberg-Marquardt algorithm iteratively includes the following steps: Step 7.1: Initial guess: Use Taylor expansion or linear interpolation results as initial values, initial slack variables: Set Step 7.2: Residual vector r = [r1, r2…, r N ] T For parameters p = [a, b, c, t] T The Jacobian matrix J is: among them k =sinθ k -(aθ+bθ+c); Step 7.3: Iteratively update the parameter p: p (n+1) =p (n) -(J T J+λdiag(J T J)) -1 J T rAdjust the damping factor. Step 7.4: Terminate the iteration when any of the following conditions are met: (1) Residual change rate |Δr| / |r|<ε; (2) parameter change rate |Δp| / |p|<ε; (3) Reach the maximum number of iterations.

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