### Cordic sin cos vhdl

Both of these approaches used only a minimum number of clocks, although their precision was somewhat limited. These rotation matrices can be strung together to accomplish many digital logic purposes.

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A simple two-dimensional rotation matrix is given by:. We could apply this rotation using angles more suited for an FPGA. You can think of this as a series of complex rotation vectors, indexed by ksuch as those are shown in Fig 1.

Notice from the figure that these vectors are not on the unit circle, but rather just outside the unit circle, and they get closer and closer to the unit circle the higher k becomes.

In other words, T is approximately a rotation matrix. T is also something that is easy to calculate within an FPGA. It requires only adds, subtracts, and shifts. Of course, this transform is not a true rotation matrix. Instead, it is a scaled rotation matrix. To see this, first calculate the angles of the vectors in Fig 1 above:. Then also calculate and normalize by their their lengths.

The resulting transform, Tis shown below:. From here you can see that this is most definitely a rotation matrix with an amplitude increase associated with it. Further, as you may have guessed from Fig 1 above, we can apply a similar rotation going in the opposite direction:.

You would then have a vector that has been rotated by Further, the more of these rotation matrices you string together, the smaller the remaining rotation becomes, and hence the closer the result will come in angular distance to any desired rotation.

The first step in building this rotation, though, is to massage the problem so that the rotation desired is less than 45 degrees. Angles beyond 45 degrees just get smaller.

Our first problem, therefore is going to be rotating our incoming vector so that any remaining rotation amount is 45 degrees or less. Because the CORDIC algorithm will also increase the magnitude of the input, this process adds one more bit on the leftâ€”to allow for a touch of width expansion.The floating point numbers are represented as integers. Both the sine and the cosine of the input angle will be computed.

The interface of the module looks as follows:. We will first write a unit test for the design. The idea is to use the cos and sin functions from the math module to compute the expected results on a number of input angles, and to compare them with the outputs from the design under test. Here is the code:.

As the design will perform its calculations in finite precision, there will be rounding errors. Therefore, the comparisons between expected and actual results are performed using an error margin.

### Compute Sine and Cosine Using CORDIC Rotation Kernel

We may want to use the test bench for designs with different characteristics; therefore, the error margin is made a parameter. The precision is specified in terms of the number of bits after the point, using the parameter fractionSize. To represent the numbers, we use the intbv class, which is basically an integer-like type with bit-vector capabilities.

Note that we constrain the intbv instances by specifying the range of valid integer values, not by a bit width. For high-level, algorithmic work, this is much easier. Moreover, it enables fine-grained range error checking at run-time.

By filling in the parameters of the bench function, we can construct an actual test bench that runs a simulation, as follows:. To implement the design, we will use the Cordic algorithm, a very popular algorithm to compute trigonometric functions in hardware.

On this page, we are mainly interested in the mechanical characteristics of the algorithm and their hardware implications. For more information and background on the algorithm itself, please consult other sources, such as this paper by Ray Andraka. The Cordic algorithm is an iterative algorithm based on vector rotations over elementary angles.

The algorithm normally operates in one of two modes.

In rotation mode, it rotates a vector x 0y 0 in the Cartesian plane over an input angle z 0. The Cordic equations for this mode are:. These equations can be implemented with relatively simple hardware. This is the characteristic that makes the Cordic algorithm attractive.The process of using the Xilinx Core Generator tool will be discussed. In a previous article, we saw that VHDL components allow us to have a neat hierarchical design and reuse a previously developed code segment several times.

We can also use this capability of hardware description languages to add optimized code segments which are developed by experienced engineers to our design. Such a ready-made component is called a core. You can find cores for a wide variety of functions, such as multipliers, digital filters, DSP-related transforms, memories and more. As a result, you can concentrate on the rest of your design and finish your project much faster.

The GUI shows a symbol for the core where you can see the active inputs and outputs of the core. Note that, based on the chosen features of the core, a terminal may or may not be active. For more details, please read the datasheet. Since the numbers are signed, the leftmost bit is the sign bit. You can find examples of the different rounding modes in the core datasheet. The third page of the settings is shown in Figure 5. According to the datasheet, setting the value of these two options to zero will force the software to automatically figure out these two parameters based on the accuracy of the outputs and other parameters.

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The last part specifies the optional pins of the core. Now, you can use this component just as you use a normal VHDL component. This simulation uses a set of waveforms similar to that of Figure 6. You can find optimized and verified cores for a wide variety of functions, such as multipliers, digital filters, DSP-related transforms, memories and more. These ready-made cores allow you to approach a large design much more easily and finish your project much faster. To see a complete list of my articles, please visit this page.

Don't have an AAC account? Create one now. Forgot your password? Click here. Latest Projects Education. Use Code Segments developed by Sophisticated Engineers In a previous article, we saw that VHDL components allow us to have a neat hierarchical design and reuse a previously developed code segment several times.

Conclusion You can find optimized and verified cores for a wide variety of functions, such as multipliers, digital filters, DSP-related transforms, memories and more. Learn More About: fpga digital signal processing digital design xilinx cpld cordic algorithm.

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Continue to site.Documentation Help Center. CORDIC-based algorithms are critical to many embedded applications, including motor controls, navigation, signal processing, and wireless communications. The Givens rotation-based CORDIC algorithm see [1,2] is one of the most hardware efficient algorithms because it only requires iterative shift-add operations. The CORDIC algorithm eliminates the need for explicit multipliers, and is suitable for calculating a variety of functions, such as sine, cosine, arcsine, arccosine, arctangent, vector magnitude, divide, square root, hyperbolic and logarithmic functions.

You can use a CORDIC rotation computing mode algorithm to calculate sine and cosine simultaneously, compute polar-to-cartesian conversions, and for other operations. In the rotation mode, the vector magnitude and an angle of rotation are known and the coordinate X-Y components are computed after rotation. Next, the rotation decision at each CORDIC iteration is done in a way that decreases the magnitude of the residual angle accumulator.

The rotation decision is based on the sign of the residual angle in the angle accumulator after each iteration. This provides the following result as approaches :. Typically is chosen to be a large-enough constant value. Thus, may be pre-computed.

To support angles outside of that range, quadrant correction is often used. This same code can be used for both fixed-point and floating-point operation.

### Cordic Algorithm Using VHDL

The judicious choice of initial values allows the CORDIC kernel rotation mode algorithm to directly compute both sine and cosine simultaneously. The angle input look-up table inpLUT is set to atan 2. After iterations, these initial values lead to the following outputs as approaches :. Other rotation-kernel-based function approximations are possible via pre- and post-processing and using other initial conditions see [1,2].

You can typically achieve greater accuracy by increasing the total number of iterations. Suppose that you have a rotation angle sensor e. Also suppose that you have a bit integer arithmetic unit that can perform add, subtract, shift, and memory operations. With such a device, you could implement the CORDIC rotation kernel to efficiently compute cosine and sine equivalently, cartesian X and Y coordinates from the sensor angle values, without the use of multiplies or large lookup tables.

Jack E. A modified version of this example exists on your system. Do you want to open this version instead? Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance. Other MathWorks country sites are not optimized for visits from your location.

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Get trial now. Toggle Main Navigation. Search Support Support MathWorks. Search MathWorks. Off-Canvas Navigation Menu Toggle. The fixed-point CORDIC algorithm requires the following operations: 1 table lookup per iteration 2 shifts per iteration 3 additions per iteration. No, overwrite the modified version Yes. Select a Web Site Choose a web site to get translated content where available and see local events and offers.

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Thanks for your replies!The CORDIC algorithm is an iterative algorithm to evaluate many mathematical functions, such as trigonometrically functions, hyperbolic functions and planar rotations. As the name suggests the CORDIC algorithm was developed for rotating coordinates, a piece of hardware for doing real-time navigational computations in the 's.

Suppose we want to rotate a point X,Y by an angle Z. The P n factor can be eliminated from the equations by pre-computing its final result. If we multiply all P n 's together we get the aggregate constant. Depending on the number of iterations and the number of bits used. Username: Password: Remember me Register. Overview News Downloads Bugtracker. Project maintainers Herveille, Richard.

Star 2 you like it: star it! Description The CORDIC algorithm is an iterative algorithm to evaluate many mathematical functions, such as trigonometrically functions, hyperbolic functions and planar rotations. Core Description As the name suggests the CORDIC algorithm was developed for rotating coordinates, a piece of hardware for doing real-time navigational computations in the 's.In fact this algorithm with some modifications allows to compute other functions used in typical calculator such as tangents, inverse trigonometric function, hyperbolic, exponentials, logarithms or roots.

Why not use Taylor series? Taylor series for sin and cos presented below can also be used but they require a lot of multiplications division in the formulas can be hard-wired and changed to multiplication. CORDIC is much better for low cost devices such as calculators, devices without hardware multiplier or where the limited amount of gates is required.

As we see, Taylor approximation is close to sin x. Algorithm works with radians so we need to recalculate angle if given value is provided in degrees.

To achieve precision to 10 digits, 40 iterations have to be performed. More iterations of course means better approximation.

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Iteration process seems to be complicated, but all these multiplications can be avoided. References: 1. How Hand Calculators Calculate 2. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email.