Frequency resolution of CORDIC algorithm

GNU Radio
1

Hi,

after having read several papers on the subject, I am still not able
to find the answer I am looking for. I wonder how to calculate the
frequency resolution of the CORDIC algorithm. In an earlier post to
this mailing list it was stated that the resolution is approximately
0.01 Hz. Could anyone point me to where I can find a deviation of this
result?

Regards,

Trond D.

2

On 6/28/07, Trond D. [email protected] wrote:

Hi,

after having read several papers on the subject, I am still not able
to find the answer I am looking for. I wonder how to calculate the
frequency resolution of the CORDIC algorithm. In an earlier post to
this mailing list it was stated that the resolution is approximately
0.01 Hz. Could anyone point me to where I can find a deviation of this
result?

This paper is really good for understanding the CORDIC:
AFS Directories Unavailable

It is specifically written to look at FPGA implementations, which is
nice.

As I understand it, the USRP uses the CORDIC as described in section
3.1 of that paper. A phase accumulator is used to spin the angle
around, and the modulated sin/cos or xi is the output on xo and yo
after 12 iterations of the algorithm. The value of zo should be zero,
and any error leftover should be represented on that output.

The resolution should really be how slowly you can spin the zi
component while maintaining accuracy out of the CORDIC. It may be
that with 12 iterations and 16-bit inputs 0.01 Hz is possible, whereas
more iterations or larger inputs might get better resolution, but I
suspect you’re really past the point of diminishing returns at that
point.

Is that helpful?

Brian

3

Trond D. wrote:

Hi,

after having read several papers on the subject, I am still not able
to find the answer I am looking for. I wonder how to calculate the
frequency resolution of the CORDIC algorithm. In an earlier post to
this mailing list it was stated that the resolution is approximately
0.01 Hz. Could anyone point me to where I can find a deviation of this
result?

The CORDIC deals with phase, not frequency, so there is no concept of
frequency resolution for it. The one being used in the USRP has 14 bits
of PHASE resolution.

Frequency resolution is controlled by the NCO, or phase accumulator. In
our case it has 32 bits of resolution. 64 MHz/2^32 is your frequency
resolution in Hz.

Matt

4

Trond D. wrote:

after having read several papers on the subject, I am still not able
to find the answer I am looking for. I wonder how to calculate the
frequency resolution of the CORDIC algorithm. In an earlier post to
this mailing list it was stated that the resolution is approximately
0.01 Hz. Could anyone point me to where I can find a deviation of
this result?

The “phase generator” part of the CORDIC block works by incrementing a
32-bit phase register by a fixed amount per clock cycle. The full size
of the register represents 2*PI() of phase, or one cycle of the
waveform. The user programmed phase increment per clock cycle then
represents frequency.

In the receive chain of the FPGA, the phase generator is clocked at 64
MHz. Thus, the minimum delta-frequency (a one bit change in the phase
increment register) is 64 MHz / pow(2, 32) = 0.0149 Hz.

Thus, for DC, the phase increment value is zero, for 0.0149 Hz, it is 1,
for 0.0298 Hz, it is 2, all the way up to 32 MHz, where it is pow(2,
31). You can also tune negative frequencies, where -1 creates -0.0149
Hz, etc.

The CORDIC block then uses the resulting “sawtooth” phase value to
rotate the incoming signal by that amount, resulting in complex
frequency conversion.

Did this help, or confuse?

–
Johnathan C.
Corgan Enterprises LLC
http://corganenterprises.com

5

2007/6/28, Johnathan C. [email protected]:

32-bit phase register by a fixed amount per clock cycle. The full size
31). You can also tune negative frequencies, where -1 creates -0.0149
Hz, etc.

The CORDIC block then uses the resulting “sawtooth” phase value to
rotate the incoming signal by that amount, resulting in complex
frequency conversion.

Did this help, or confuse?

Thanks a lot! This was indeed the last missing piece in my CORDIC
puzzle, which by no means covers the entire picture, but a sufficient
subset for my humble needs :slight_smile:

–
Trond D.

6

2007/6/28, Johnathan C. [email protected]:

Thus, for DC, the phase increment value is zero, for 0.0149 Hz, it is
1,
for 0.0298 Hz, it is 2, all the way up to 32 MHz, where it is pow(2,
31). You can also tune negative frequencies, where -1 creates -0.0149
Hz, etc.

There is one more thing that I just can not figure out. The largest
angular rotation that can be performed by the CORDIC is +/- pi/2,
which means that is takes four cycles to rotate the vector all the way
around. How come the largest frequency that can be generated be 32
MHz.

Thanks in advance

Regards,

Trond D.

7

2007/6/28, Brian P. [email protected]:

This paper is really good for understanding the CORDIC:
The resolution should really be how slowly you can spin the zi
component while maintaining accuracy out of the CORDIC. It may be
that with 12 iterations and 16-bit inputs 0.01 Hz is possible, whereas
more iterations or larger inputs might get better resolution, but I
suspect you’re really past the point of diminishing returns at that
point.

Is that helpful?

Thank you a lot for your reply. I have already read the mentioned
paper, and found it useful.

Regards,

Trond D.

8

Trond D. wrote:

around. How come the largest frequency that can be generated be 32
MHz.

If you look in the cordic.v file you will see that the actual CORDIC is
preceded by some logic which rotates by 0, 90, 180, or 270 degrees.

Matt