New Project: libhackrf Racket bindings

I’m fond of my HackRF radio, but lost interest in working through GnuRadio Companion, because I don’t want to do any Python programming to extend the blocks. So I started looking into scheme bindings for libhackrf. There is a learning curve, for sure, but it appears that this shouldn’t be too difficult, using the Racket FFI. Here is some code (minus some important error checking) that pulls the hackrf_device_list, i.e., the struct containing information about what HackRF devices are currently plugged in to your USB:

#lang racket/base

(require ffi/unsafe
         ffi/unsafe/define)
 
(define-ffi-definer define-hackrf (ffi-lib "libhackrf"))

(define-hackrf hackrf_init (_fun -> _int))
(define-hackrf hackrf_exit (_fun -> _int))

(define _hackrf_device_list_t-pointer (_cpointer 'hackrf_device_list_t))

(define-hackrf hackrf_device_list (_fun -> _hackrf_device_list_t-pointer))

(define-cstruct _hackrf_device_list_st ([serial_numbers _pointer]
                                        [usb_board_ids _pointer]
                                        [usb_device_index _pointer]
                                        [devicecount _int32]
                                        [usb_devices _pointer]
                                        [usb_devicecount _int32]))

This ties into the C API from hackrf.h (ADDAPI and ADDCALL are blank except in Windows):

struct hackrf_device_list {
	char **serial_numbers;
	enum hackrf_usb_board_id *usb_board_ids;
	int *usb_device_index;
	int devicecount;
	
	void **usb_devices;
	int usb_devicecount;
};
typedef struct hackrf_device_list hackrf_device_list_t;

extern ADDAPI int ADDCALL hackrf_init();
extern ADDAPI int ADDCALL hackrf_exit();

extern ADDAPI hackrf_device_list_t* ADDCALL hackrf_device_list();

And here is a demo function to get the device count:

(define (hackrf-device-count)
  (hackrf_device_list_st-devicecount
   (ptr-ref (hackrf_device_list)
    _hackrf_device_list_st)))

With my HackRF plugged in:

racket@hackrf-ffi.rkt> (hackrf_init)
 0
 racket@hackrf-ffi.rkt> (hackrf-device-count)
 1

After unplugging it:

racket@hackrf-ffi.rkt> (hackrf-device-count)
0

Elliptic Pencil

I wanted to play around with generating an “elliptic pencil of circles” starting with the Hermatian matrices. An elliptic pencil of circles are all the circles (or some of them) which all intersect at two specific points. Here is a simple one, where the points of intersection are at (0,1) and (0,-1), or ±i in the complex plane.

You see there is the unit circle, plus the other circles branching off to each side:

The Hermatian matrix template for this particular pencil is elegant:

\begin{bmatrix} 1 & -n \\ -n & -1 \end{bmatrix}

where n is the x coordinate of the center of each circle (or the complex number center, not having an imaginary component).

Here is Racket code used to generate the plot:

(require math/array)
(require math/matrix)
(require plot)

(define (matrix-circle-radius M)
  (let ([A (array-ref M #[0 0])]
        [d (matrix-determinant M)])
    (sqrt (/ d (* -1 (* A A))))))

(define (matrix-circle-center M)
  (let ([C (array-ref M #[1 0])]
        [A (array-ref M #[0 0])])
    (/ C (* -1 A))))

(define (circle-isoline x y r c)
  (isoline
   (lambda (x_ y_) (sqrt (+ (sqr (- x_ x)) (sqr (- y_ y))))) r
   #:color c
   #:width 2))

(define (elipt-plot-demo)
  (plot
   (map (lambda (C)
          (circle-isoline (real-part (first C))
                          (imag-part (first C))
                          (second C)
                          (list 30 160 210)
                          ))
        (map (lambda (M)
               (list (matrix-circle-center M)
                     (matrix-circle-radius M)))
             (map (lambda (n)
                    (matrix+
                     (matrix-scale (array #[#[0 1] #[1 0]]) n)
                     (array #[#[1 0] #[0 -1]])))
                  (range 2.5 -3 -0.5))))
                    
   #:x-min -5
   #:x-max 5
   #:y-min -5
   #:y-max 5
   #:width 400
   #:height 400))

Circles Common Invariant

Schwerdteger describes an interesting “invariant” relationship between any two circles:

\frac{\Delta_{12}}{\sqrt{\Delta_1} \sqrt{\Delta_2}}

where

\Delta_1 = | \mathfrak{C}_1 |, \Delta_2 = | \mathfrak{C}_2 |, and 2 \Delta_{12} = A_1 D_2 + A_2 D_1 - B_1 C_2 - B_2 C_1

with the circles represented as Hermitian matrices \begin{bmatrix} A & B \\ C & D \end{bmatrix}.

The point is, in the end you come up with this single number which represents whether the smaller circle is inside the first, overlapping the first, just touching the first, or outside the first. Here are the cases:

common invariant > 1 : smaller circle contained withing the greater circle

common invariant = 1 : touching from the inside

-1 > common invariant > 1 : overlapping at two points

common invariant = -1 : touching from the outside

common invariant < -1 : completely outside

Here is Racket code used for calculations and plotting:

(require math/array)
(require math/matrix)
(require plot)

(define (circle-to-matrix zC r)
  (let ([B (* -1 (conjugate zC))]
        [mzC (magnitude zC)])
    (matrix [[ 1             B                       ]
             [ (conjugate B) (- (* mzC mzC) (* r r)) ]])))

;; gothic C - 212d
;; delta - 394
(define (invariant ℭ1 ℭ2)
  (let* ([Δ1 (matrix-determinant ℭ1)]
         [Δ2 (matrix-determinant ℭ2)]
         [A1 (array-ref ℭ1 #[0 0])]
         [B1 (array-ref ℭ1 #[0 1])]
         [C1 (array-ref ℭ1 #[1 0])]
         [D1 (array-ref ℭ1 #[1 1])]
         [A2 (array-ref ℭ2 #[0 0])]
         [B2 (array-ref ℭ2 #[0 1])]
         [C2 (array-ref ℭ2 #[1 0])]
         [D2 (array-ref ℭ2 #[1 1])]
         [Δ12 (* 0.5 (+ (* A1 D2)
                        (* A2 D1)
                        (* -1 B1 C2)
                        (* -1 B2 C1)))])
    (/ Δ12 (* (sqrt Δ1) (sqrt Δ2)))))

(define (circle-isoline x y r)
  (isoline
   (lambda (x_ y_) (sqrt (+ (sqr (- x_ x)) (sqr (- y_ y))))) r))

(define (circles-invariant-plot x1 y1 r1 x2 y2 r2 min max)
  (let ([C1 (circle-to-matrix (make-rectangular x1 y1) r1)]
        [C2 (circle-to-matrix (make-rectangular x2 y2) r2)])
    (plot-file
     (list
      (circle-isoline x1 y1 r1)
      (circle-isoline x2 y2 r2)
      (point-label (vector (+ min (* (- max min) 0.1))
                           (+ min (* (- max min) 0.9)))
                   (number->string (invariant C1 C2)) #:point-size 0))
     "out.png"
     #:x-min min
     #:x-max max
     #:y-min min
     #:y-max max
     #:width 400
     #:height 400)))

Off-On Fourier Series

I have been fascinated lately with the concept of frequency spectrum and the idea that all periodic signals can be approximated by an infinite sum of sinusoidal functions. There are many introductory videos on this subject, usually titled as introductions to the Fourier transform.

As far as the actual math involved, this YouTube* video was very helpful:
Compute Fourier Series Representation of a Function

I don’t actually use the YouTube Website directly because of the massive amounts of proprietary JavaScript involved, but instead use youtube-dl to download the video.)

He converts an off-on type of function to a fourier series. After the integration, we get this:

I translated that into some plots in Racket, to give the visual idea. Say we only add in a single sinusoid:

Then, another:

And a few more:

And a lot more:

And finally, hundreds of them:

It cannot quite perfectly represent the function, because the Fourier series adds an extra point in between the switch from off to on (and back), whereas the original just jumps from 0 to 1 (and back).

Here is the Racket code for those interested (I did not bother to optimize):

#lang racket

(require plot)

(define (pulse x l)
  (letrec ([pulse_ 
            (lambda (acc n)
              (if (> n l) acc
                  (pulse_
                   (+ acc
                      (/ (* 2
                            (sin (* (+ (* 2 n) 1) pi x)))
                         (* (+ (* 2 n) 1) pi)))
                      (+ n 1))))])
    (pulse_ 0.5 0)))
        
(define (pulseplot l)
  (plot
   (function (lambda (x) (pulse x l)) -1 3)))

Representation of Circles by Hermitian Matrices

I borrowed the book Geometry of Complex Numbers by Schwerdtfeger. The material has been fascinating so far, though admittedly it took me about 3 hours to comprehend the material in page 1 of chapter 1. The first subject is this intriguing idea that you can represent a circle as a matrix. More specifically, you you can represent a circle with (complex) center 𝛾 and radius 𝜌 as a matrix

\begin{bmatrix} A & B \\ C & D \end{bmatrix}

Where B = – A𝛾̅, C = -A𝛾, and D = A(𝛾𝛾̅-𝜌²), so that A and D are real numbers, and B and C are complex numbers. For normal circles, you will have A=1, though you can scale the matrix to have other (non-zero) values of A and still have the same circle. (If A is zero, you end up with a straight line, or some other mysterious thing that is not a circle.)

The simple case is the unit circle:

Which is \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}

More exotic is the circle away from the origin, centered at 10+10i, i.e., (10, 10), with radius 3:

\begin{bmatrix} 1 & -10+10i \\ -10-10i & 191 \end{bmatrix}

Here is some code to generate the matrix in Racket, and to put it back:

lang racket

(require math/array)
(require math/matrix)

(define (circle-to-matrix zC r)
  (let ([B (* -1 (conjugate zC))]
        [mzC (magnitude zC)])
    (matrix [[ 1             B                       ]
             [ (conjugate B) (- (* mzC mzC) (* r r)) ]])))

(define (matrix-circle-radius M)
  (let ([A (array-ref M #[0 0])]
        [d (matrix-determinant M)])
    (sqrt (/ d (* -1 (* A A))))))

(define (matrix-circle-center M)
  (let ([C (array-ref M #[1 0])]
        [A (array-ref M #[0 0])])
    (/ C (* -1 A))))
racket@circle.rkt> (circle-to-matrix 0 1)
(array #[#[1 0] #[0 -1]])
racket@circle.rkt> (matrix-circle-radius (circle-to-matrix 0 1))
1
racket@circle.rkt> (matrix-circle-center (circle-to-matrix 0 1))
0

Why is this significant? I think Schwerdtfeger is going to cover that on page 2. :)