Lecture 20 – CORDIC

References

• $^\dagger$ https://youtu.be/HjwRdsjIuVQ CORDIC design in Verilog to produce sine and cosine functions -Kirk Weedman
• Additional Reading Source: Koren, Israel. Computer arithmetic algorithms, 2nd ed. AK Peters/CRC Press, 2001.

CORDIC

• COordinate Rotation Digital Computer

• developed by Jack Volder in 1959

• developed as an iterative algorithm to convert between polar and cartesian coordinates using ONLY shift, add, and subtract operations

• Has two modes of operation (circular rotation and vectoring) that can be used to support computation of trig,hyperbolic, linear and logarithmic functions, sin/cos, tan, polar-rect coordinate conversion

• Circular rotation mode of operation:

• Using an iterative algorithm, transform a vector $x_0,y_0$ to a resulting vector $x_N,y_N$ using N iterations

  • inputs :
    • x0,y0: input vector v
    • $\theta$: angle
  • outputs :
    • xN,yN: output vector after N iterations
  • Our goal is to compute a final x,y using n iterations
    $x_n = x_0 \cos (\theta) - y_0 \sin(\theta)$
    $y_n = y_0 \cos (\theta) + x_0 \sin(\theta)$
  • Simplifying:
    Factor $cos(\theta)$:
    $x_n = \cos (\theta) (x_0 - y_0 \tan(\theta))$
    $y_n = \cos (\theta) (y_0 + x_0 \tan(\theta))$

We'll address the two multiplications of each equation to simplify them

Simplifying multiplications by $\tan(\theta)$

• we are going to restrict allowable rotations $\theta$ such that $\tan(\theta) = \pm 2^{-i}$ so that the multiplication can be replaced by a shift and add/sub. We'll predetermine allowable rotations and store them in a table:

  i	$\tan(\theta)=\pm 2^{-i}$	$\tan^{-1}(2^{-i}) = \theta_i$	$\frac{\theta_i}{2\pi}$	degrees
  0	$\pm 2^{-0}$	$\pm \tan^{-1}(1) = \pm 0.7853981633974483$	$\pm 0.125=1/8^{th}$ turn	$\pm 45.0$
  1	$\pm 2^{-1}$	$\pm \tan^{-1}(1/2) = \pm 0.4636476090008061$	$\pm 0.07379180882521663$	$\pm 26.565051177077986$
  2	$\pm 2^{-2}$	$\pm \tan^{-1}(1/4) = \pm 0.24497866312686414$	$\pm 0.03898956518868466$	$\pm 14.036243467926479$
  3	$\pm 2^{-3}$	$\pm \tan^{-1}(1/8) = \pm 0.12435499454676144$	$\pm 0.019791712080282773$	$\pm 7.125016348901799$
  4	$\pm 2^{-4}$	$\pm \tan^{-1}(1/16) = \pm 0.06241880999595735$	$\pm 0.00993426215277042$	$\pm 3.5763343749973515$
  5	$\pm 2^{-5}$	$\pm \tan^{-1}(1/32) = \pm 0.031239833430268277$	$\pm 0.004971973911794637$	$\pm 1.7899106082460694$
  6	$\pm 2^{-6}$	$\pm \tan^{-1}(1/64) = \pm 0.015623728620476831$	$\pm 0.0024865936394752068$	$\pm 0.8951737102110744$
  7	$\pm 2^{-7}$	$\pm \tan^{-1}(1/128) = \pm 0.007812341060101111$	$\pm 0.0012433726968348697$	$\pm 0.4476141708605531$
  8	$\pm 2^{-8}$	$\pm \tan^{-1} (1/256) = \pm 0.0039062301319669718$	$\pm 0.0006216958343570503$	$\pm 0.22381050036853808$
  9	$\pm 2^{-9}$	$\pm \tan^{-1}(1/512) = \pm 0.0019531225164788188$	$\pm 0.0003108491029616858$	$\pm 0.1119056770662069$
  10	$\pm 2^{-10}$	$\pm \tan^{-1}(1/1024) = \pm 0.0009765621895593195$	$\pm 0.0001554246997050102$	$\pm 0.05595289189380367$
  ...	...	...	...	...

• To achieve other angles not in the restricted set, we can allow a combination of multiple rotations from any of the allowable angles.

• The series of rotations can be first calculated using an iterative process with an angle accumulator R as follows

• Initialize R to $\theta$

• In each iteration i, we will rotate by $\theta_i$ or $-\theta_i$ according to R: $R=R-r_i \theta_i$; $r_i=\begin{cases}1,&R\ge0\\-1,&\text{otherwise}\end{cases}$

  • R $\to$ 0, the total rotation is recorded in $r_i$: $\sum_{i=0}^\infty r_i \cdot \theta_i$

Supported Input Argument Range

• The range for this approach can be found by assuming $r_i$ is the same in every iteration (+1):
  • $\frac{\sum_{i=0}^\infty \theta_i}{2 \pi} \approx 0.27745$
  • Therefore the range is just beyond $\pm$ $^1/_4$ turn ($\approx \pm 100^\circ$)

A method to increase the range will be achieved with an initialization step

Visualization of Non-Restoring CORDIC Process

Simplifying multiplications by $\cos (\pm\theta_i)$

Iteration:

• $x_{i+1} = \cos (\pm\theta_i) (x_i - y_i \tan(\pm\theta_i))$

• $y_{i+1} = \cos (\pm\theta_i) (y_i + x_i \tan(\pm\theta_i))$

• Note:
  $\cos (\textcolor{red}{\pm}\theta_i) = \textcolor{red}{+}\cos (\theta_i)$ ;
  $\tan (\textcolor{red}{\pm}\theta_i) = \textcolor{red}{\pm}\tan (\theta_i)$

• therefore

  • $x_{i+1} = \cos (\theta_i) (x_0 - \red{r_i} \cdot y_i tan(\theta_i))$
  • $y_{i+1} = \cos (\theta_i) (y_0 + \red{r_i} \cdot x_i tan(\theta_i))$

• We can pre-calculate each cos as ($K_i$) and use the fact that $\tan(\theta_i)=2^i$

  $K_i = \cos(\tan^{-1}\left(2^{-i}\right))=\frac{1}{\sqrt{1+2^{-2i}}}$
  
  
  $X_{i+1} = K_i [X_i-r_i Y_i 2^{-i}]=K_i [X_i-r_i (Y_i \texttt{>>}i)]$
  $Y_{i+1} = K_i [Y_i+r_i X_i 2^{-i}] =K_i [Y_i+r_i (X_i \texttt{>>}i)]$

• Furthermore we can save all the multiplications by $K_i$ till the end and multiply just the final result by $\prod K_i$

• Hence, each stage is reduced to a shift and an add/sub for X and Y !!!!

Number of Required Iterations

• If the output is n bits, then an iteration with i=n would involve a shift of n producing an add/sub 0, no update.
  $X_{n+1}=X_n-r_n \red{(Y_i \texttt{<<}n)} = X_n-r_i \cdot \red{\cancel0} = X_n$
• Perform iterations until but not including i=n

Initialization Step to get angle within $\frac{2\pi}{4}$ from 0

Introduce initial rotation to reduce |R| to be less than $\frac{2\pi}{4}$

• May note that tan is unique only an interval $\frac{2\pi}{4} \lt \theta \lt -\frac{2\pi}{4}$ (Q1 or Q4), but this doesn't turn out to be an issue in the meathod we'll use ...

• We'll use an alternative, cheap initial rotation without use of trigonometric functions.

• While only Q2 and Q3 rotation vectors are a problem, Volder proposed to rotate all inputs by $\pm \pi /2$

• • $X_0 = - v \cdot Y$
  • $Y_0 = v \cdot X_0$
  • $\theta_0 = \theta -v \cdot \frac{\pi}{2}$
  • $v = \begin{cases} +1,&\theta_0>0\\ -1 & {\rm otherwise} \end{cases}$
• • Note that Q1 vectors are moved into Q4 and visa versa, which is fine.

• Example

  • presented with $170^\circ$
  • mirroring (rotation by $180^\circ$) could be implemented by $X_0=-X$,$Y_0=-Y$ resulting in a remaining $-10^\circ$ rotation to perform
  • Volder's solution instead suggests to note that the rotation vector is in the upper quadrants and therefrom perform an initial rotation by $90^\circ$ (subtracting $90^\circ$ from the remaining rotation) by modifying the provided coordinates: $X_0=-Y$,$Y_0=X$

• Example (figure provided)
  • presented with $120^\circ$ and X=1.4,Y=.2
  • Volder's solution: since rotation%360 is in range 0 to 180, perform an initial rotation by $90^\circ$, by the action on the coordinates $X_0=-Y$, $Y_0=X$

Number of Steps Required (Complexity/Convergence)

• n+1 iterations including initial iteration:

  • initial iteration: check sign, swap, one negate
  • n iterations: check sign, shift-add/sub

• final scale by $\prod K_i$ can be performed if desired (converges to approximately a factor of 1.647 for a few iterations or more )

  • can also treat this as "system gain" factor and combine with overall system gain
    • system gain factors are common in DSP operations
    • correction at end may or may not be required based on the application. If comparing to other results with the same system gain (scale factor), the adjustment is not nessiarliy.

Speed and Pipelining

• CORDIC requires n+1 iterations

  • single hardware iteration stage

    • if run at system clock, computation suffers latency and low throughput
    • if using run at a fast clock, multi-clock-domain design must be introduced

  • pipelining allows throughput to "match" system clock rate (still have latency)

    • requires hardware per iteration, typically more area
    • stages are simplified as they are specialized for each iteration. LUT for the allowed $\theta_i$ is replaced by a hardwired constant per stage

Computation of SINE and COSINE

• computed using a special case of vector rotation
• initialize system with a specific vector: X=1, Y=0
• final result:
  • X:$\cos(\theta)$
  • Y:$\sin(\theta)$

Free multiplication

• initialize system with X=A, Y=0, $\theta$

  • final result:
    • X:$A \cdot \cos(\theta)$ ; Y:$A \cdot \sin(\theta)$

• initialize a pipeline implementation at each time t:
  $X=A(t)$; $Y=0$; $\theta(t)$=$\frac{F}{f_{clock}}\times2\pi\times t$

  • result:
    • X(t):$A(t-(m+1))\cos(\theta(t-(m+1)))$
    • Y(t):$A(t-(m+1))\sin(\theta(t-(m+1)))$
  • This is a modulator with a latency (m+1)

Designing N-BIT REPRESENTATION of $\theta$

If mapping the full binary range of n bits to the range of angles around the circle, let 0000 be angle of 0 and 1111...1 be the last angle (adjacent and clock-wise of 0)

• angle resolution is $\frac{2\pi}{2^n}$
• the first two bits determine the quadrant
• two's complement or unsigned inputs can be accepted without modification
• any modular arithmetic naturally maps to the correct result (mod $2^n$ :: mod $360^\circ$) e.g.
  $180^\circ+180^\circ\to0$ ; $90^\circ\pm180\to-90^\circ$

CORDIC Pipelining and Applications

• CORDIC is naturally pipelined (upper right), accepting new initial values for X,Y and $\theta$ every cycle

  

• Applications:

  • With fixed input X,Y and incrementing $\theta$ by the appropriate rate $\Delta\theta=f/(2\pi)/\text{sample rate}$, sinusoidal outputs are produced. An offset of theta determines a phase offset for the output sinusoids
  • Furthermore, varying the inputs according to a time varying-input signal produces a modulation of the that input signal, such as for frequency shifting
  • Varying $\Delta\theta$ over time according to a second input can produce frequency modulation