In this article, we will briefly discuss the Goertzel algorithm flow and its fixed-point implementation on the Analog Devices Blackfin BF5xx processor family using the Blackfin's special arithmetic modes. In particular, we will discuss an efficient implementation of 16.16 fixed-point multiplications on the 16-bit MAC-friendly Blackfin DSP processor.
Goertzel Algorithm
The Goertzel algorithm is widely used for the detection of a few frequencies in a given signal input.
Though an N-point Fast Fourier Transform (FFT) algorithm efficiently computes N Discrete Fourier Transform (DFT) coefficients given the N input samples, some applications, such as dual tone multi-frequency (DTMF) don't require all the DFT coefficients. In such cases, the Goertzel algorithm can be used to compute the required DFT coefficients (or frequencies) of the input signal x[n] efficiently using a second order linear filter.
In [Pro1992], a recursive expression for computing kth DFT coefficient is derived from the standard DFT equation as:
Now, the Z domain transfer function governing the above difference equation can be expressed as:
The final expressions for implementation of the Goertzel algorithm are obtained from the above system functions as below:
From Equation (1),
From Equation (4), we are only interested in the infinite impulse response (IIR) filter output to compute the DFT coefficient. Thus, we need not compute the finite impulse response (FIR) filter Yk[z]/Sk[z] output for each input sample sk.
The magnitude square value of kth DFT coefficient can be obtained from Equation (4) as:
The computation sk for using Equation (3) requires one multiply and two add operations per sample. In applications like DTMF detection, we are only concerned with the magnitude square of the kth DFT coefficient, i.e. |X[k]2, as given in Equation (5).
Unlike DFT, which requires storing many twiddle factors, we need not store any coefficients in the Goertzel algorithm. We use only one coefficient (q = 2cos(2πk/N) ) in the computation and this can be held using a single data register.
