Step 7: Optional: Algorithm Details

Picture of Optional: Algorithm Details
Response without Window.gif
Response with Hamming Window.gif
Hamming Window.gif
The Goertzel algorithm is a signal processing algorithm which is used for detecting a single frequency. It is derived from the Fourier transform.

The algorithm acts as a very narrow band pass filter. It produces a very sharp response to frequencies within the pass band, and a much lower response for frequencies outside the pass band. The minute details of the Goertzel algorithm won’t be covered here.  

The algorithm loop samples the input using the microprocessor built in A/D converter. The necessary mathematical operations are performed in the time between successive samples.
The different variables used in the algorithm are defined as follows:

Xn_sample=The latest sample from the A/D converter
Xn=The latest sample from the A/D converter multiplied by the window function.
Y0=The output value presently being computed.
Y1=The previously computed output value. (The output value from the previous iteration of the loop).
Y2=The output value from the iteration the iteration before the l previous iteration of the loop.

The figure below shows a list of the variables used in the algorithm.

Refer to the flow chart in the attached PDF to see how the Goertzel algorithm portion of the program executes on the processor. Once the algorithm has processes all the samples (the attached *.hex and *.asm files use 200 samples), the real and imaginary components of the results are calculated. The real and imaginary portions are then used to compute the magnitude. The magnitude is a measure of the target frequency is present in the sampled data. The magnitude is then compared against the threshold. If the value is greater than the threshold value, the frequency is considered to be detected, and the output is set HIGH. If the magnitude is lower than the threshold, the output is set LOW.

The sharpness of the filter response versus frequency is proportional to the number of samples taken. The response of the algorithm must be sharp enough that it responds to the target frequency but produces much lower response for frequencies outside the target. A value of 200 samples was found to produce a reasonably narrow response in experimentation.

Windowing Function

After each sample is obtained from the A/D converter, it is first multiplied by a window value before being used in the subsequent calculations. Refer to the pictures below showing the frequency response of the algorithm to see the affects of the window. The magnitude of the response is lower and the width of the response is wider when the window is used, but ripples are nearly eliminated, making the determination of whether the target frequency is present easier.

There are many types of windows that can be applied in digital signal processing. This particular one is called a Hamming window. The shape of the hamming window is shown in a picture below. The plot below shows the value the window function that is used for each of the samples in the subroutine.