Applications Of Goertzel Algorithm

8/10/2019 Applications Of Goertzel Algorithm

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Applications Of

Goertzel Algorithm

Presented by

Wattamwar Reshma R.

Roll No:13

ME-FY(Digital system)


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What is Goertzel algorithm?

The Goertzel algorithmis a Digital Signal

Processing technique that provides a means for

efficient evaluation of individual terms of theDiscrete Fourier Transform.


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A basic Goertzel

Before you can do the actual Goertzel, you must

do some preliminary calculations:

1. Decide on the sampling rate.

2. Choose the block size, N.

3. Precompute one cosine and one sine term.

4. Precompute one coefficient.


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Sampling rate:

The number of samples carried per second is

called sample rate.

Your sampling rate may already be determinedby the application. For example, in telecom

applications, it's common to use a sampling rate

of 8kHz (8,000 samples per second).


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Block size:

Goertzel block size N is like the number of points

in an equivalent FFT. It controls the frequency

resolution (also called bin width).

For example, if your sampling rate is 8kHz and Nis 100 samples, then your bin width is 80Hz.


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Precomputed constants

per-sample processing

calculate Q0, Q1, and Q2.

Q0 = coeff * Q1 - Q2 + sample

Q2 = Q1

Q1 = Q0

After running the per-sample equations N times,

it's time to see if the tone is present or not.

s

tone

f

fNk


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Magnitude calculation:

real = (Q1 - Q2 * cosine)

imag = (Q2 * sine) (Magnitude)2 = (real)2+(imag)2


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An optimized Goertzel

The optimized Goertzel requires lesscomputation than the basic one, at the expense

of phase information.

The per-sample processing is the same, but the

end of block processing is different. Instead of

computing real and imaginary components, and

then converting those into relative magnitude

squared, you directly compute the following:

(Magnitude)2=(Q1)2+(Q2)2-Q1*Q2*coeff


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Implementation

(1) The first stage calculates an intermediate sequenceS(n).

(2) The second stage applies the following filter to S(n),

producing output sequence Y(n).


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Implementation

The Z transform of the first filter stage given in equation

The Z transform of the second filter stage given in equation

The combined transfer function of the cascade of the two filter stagesis then


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Applications

Dual Tone Multi-Frequency (DTMF) applications.

Push-button digital telephone sets.

electronic mail and telephone banking systems.

Tuning of guitar string.

Detection of specific frequencies in audio signals.


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DTMF Signaling

In a DTMF signaling system a combination of

two frequency tones represents a specific digit,

character (A, B, C or D) or symbol (* or #).

Two types of signal processing are involved: Coding or generation.

Decoding or detection.

For coding, two sinusoidal sequences of finitelength are added in order to represent a digit,

character or symbol as shown in the following

example.


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DTMF Tone Generation

1 2 3

654

7 8 9

#0*

A

B

C

D

1209Hz 1336Hz 1477Hz 1633Hz

697Hz

770Hz

852Hz

941Hz

1336770770770770

Ou

tput

Freq (Hz)

Example: Button 5 results in a 770Hz and a

1336Hz tone being generated simultaneously.


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N = 205 fs = 8kHz The decimal values are divided by 2 to be represented in

Q15 format. This has to be taken into account during implementation

Frequency kCoefficient

(decimal)

Coefficient

(Q15)

1633 42 0.559454 0x479C

1477 38 0.790074 0x6521

1336 34 1.008835 0x4090*

1209 31 1.163138 0x4A70*

941 24 1.482867 0x5EE7*852 22 1.562297 0x63FC*

770 20 1.635585 0x68B1*

697 18 1.703275 0x6D02*


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DTMF Tone Detection

Detection of tones can be achieved by using abank of filters or using the Discrete Fourier

Transform (DFT or FFT).

However, the Goertzel algorithm is more efficient

for this application. The Goertzel algorithm is derived from the DFT

and exploits the periodicity of the phase factor,

exp(-j*2k/N) to reduce the computational

complexity associated with the DFT, as the FFT

does.


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With the Goertzel algorithm only 16 samples of

the DFT are required for the 16 tones


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Advantage

Simple structure of algorithm makes it well

suited to small processors.

More efficient than FFT for small number of

frequencies ( if M < log2N).


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THANK YOU!

Applications Of Goertzel Algorithm

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