half-band-filter

An implementation of a half-band FIR filter, from MATLAB to fixed point in SystemVerilog. It annoyed me there was not much fixed point FIR filter code, showing how to take a Matlab model and generate RTL.

Contents

• readme.md you're reading this
• ./matlab/ - Directory containing Matlab functions to design the half-band filter, check the implementation in fixed point, and create test vectors for RTL vector matching
• ./rtl/ - Directory containing SystemVerilog RTL code consisting of a synthesizable module and a test bench using the test vectors created with Matlab.
• ./images/ - images for this repo

Half-Band Filters

Check out the ./matlab/ directory for more information on half-band filters. The impulse response for the 27-tap half-band filter is shown below:

We can see from the impulse response that this is an odd-length Symmetric FIR filter which approximates a windowed discrete Sinc function, and every 2nd tap except for the middle tap has a value of 0, and the taps have symmetry about the center tap. This allows the filter to be implemented in an efficient manner:

only 8 multiplies are required to implement this 27-tap filter.

Matlab Implementation

fi, and the problem with fi

In RTL (Register Transfer Level) designs, fixed-point filters are commonly implemented using different formats, with Q Format being the most prevalent. However, there are other formats used in the industry as well. While the MATLAB fi functions are valuable for checking overflow and optimizing bit-widths, the actual filter implementation typically employs the Q format. This creates a discrepancy between MATLAB and the implemented designFortunately, converting from the Matlab floating point to Q format is very straightforward, as you can see below.

Q Format and Q Conversion to/from float:

For this code, Q15 is used and consists of a sign bit plus 15 fractional bits = 16 bits. For more information on Q Format, see the Wikipedia page.

For our purposes, Q conversions are as follows:

Float to Q

To convert a number from floating point to Qm.n format:

Multiply the floating point number by 2^n. Round to the nearest integer. Note that in software, it is common to convert float to Q by multiplying by (2^n -1). In hardware, 2^n is more common. 0.5 would be exactly represented as 2^14 in this case, whereas there would be a very small error in the case of 2^n -1.

Q to float

To convert a number from Qm.n format to floating point:

Convert the number to floating point as if it were an integer, remove the binary point Multiply by 2^-n.