## Introduction: Reliable Frequency Detection Using DSP Techniques

Accurate Frequency Detection is important for many projects such as Guitar/Piano Tuners, Vibration Analyzers, Heartrate Monitors, MEMs Sensor Analysis and Laboratory Instruments.

There have been many fine examples of projects that try to solve this problem, for example: Arduino Frequency Detection by amandaghassaei and Arduino Frequency Counter Library.

But they all use Time Domain techniques; analyzing the signal for features such as : Zero-Crossings, Peak Detection, Slope Detection etc..

Take a look at the Waveforms shown. One of them is recorded from a Piano playing Middle-C (C4) . The other is from a Synthesizer Playing Middle-C (C4). Clearly any good Time Domain algorithm will work well with the Piano waveform. But the Synthesizer waveform will not be identifiable that way because its very strong harmonic content makes the fundamental frequency undiscernable . It looks impossible to Identify the Frequency of this signal.

It is possible.

Using the technique I'm going to show you it was measured to be **259.91Hz **... only 0.09Hz away from an Exact Middle C Frequency of 260Hz.

**UPDATE**:

I got a lot of questions about how to use this to make a Guitar Tuner. SoI spent some time writing a tutorial on how to make a Guitar Tuner using this method. You can find it on my blog:

http://www.akellyirl.com/arduino-guitar-tuner/

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By the way. If you like this Instructable, you might also like to read My Blog covering various projects and Tutorials.

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## Step 1: You Will Need

But the principles apply to any platform.

To demonstrate the principles I'm going to use pre-recorded sound files captured as an array in a .h file. So we won't be needing any circuit for the Arduino this time.

## Step 2: Autocorrelation

What we need to do is to change the original signal into another one that highlights the periodicity of the original signal. So if it is indeed periodic, then that will stand out in the new signal and then we can measure that in the usual way using peak-detect or zero crossing detect.

What's the magic algorithm that does that?

It's

**Autocorrelation**.

Imagine your signal is contained in a window or buffer. Now image you have an exact copy of that window or buffer with a time delay.

What Autocorrelation does is to measure the correlation (or similarity) between the signal and its delayed copy each time the copy is delayed by a sample period.

See the diagram. When the signal and the copy have no delay they are very similar (i.e. highly correlated) as shown in step 1, and therefore the autocorrelation value for delay = 0 is maximum.

Step 2 shows that when the copy is delayed significantly it doesn't look similar to the original in the overlapping area. Therefore the autocorrelation value for this delay is small.

Step 3 shows that when the copy is delayed even more the signal in the overlapping area is very similar to the original because the signal is periodic. Therefore the autocorrelation value for this delay shows a peak.

We can see that the distance in time between the maximum peak at the beginning and the first peak afterwards must be equal to the fundamental period of the waveform.

Now that we've emphasised the periodicity of the signal by Autocorrelation we just need to perform a Peak-Detect to measure the period.

Technically the "similarity" or correlation between the signal and its delayed copy is the

**sum of the product**of the two signals.

For the technically minded all the details of Autocorrelation can be found here: http://en.wikipedia.org/wiki/Autocorrelation

## Step 3: Autocorrleation Code

for(i=0; i < len; i++) { sum = 0; for(k=0; k < len-i; k++) sum += (rawData[k]-128)*(rawData[k+i]-128)/256; }The data is in the

**rawData[]**array. We subtract 128 from each value because it's 8bit unsigned and we require signed values.

The

**sum**value is the result of each autocorrelation calculation, i.e. each point of the function. In order to save memory we don't save the output to an array. We're going to work on the individual

**sum**values to find the first peak and therefore calculate the period.

Sending the sum values out to be plotted, we get the Autocorrelation function shown. Comparing to the original signal (also shown) it is clear that there is periodicity in the original signal and this has been clearly highlighted by the Autocorrelation function.

## Step 4: Peak Detect

// Peak Detect State Machine if (pd_state == 2 && (sum-sum_old) <=0) { period = i; pd_state = 3; } if (pd_state == 1 && (sum > thresh) && (sum-sum_old) > 0) pd_state = 2; if (!i && pd_state == 0) { thresh = sum * 0.5; pd_state = 1; }The state machine moves from one state to the next when an event occurs as follows:

STATE0 : Set

**thresh**the threshold under which value we'll ignore the data : NEW STATE = 1

STATE1 : look for the signal being above the threshold AND the slope of the signal is positive : NEW STATE = 2

STATE2 : look for the slope of the signal is negative or zero. If so we've

**found the PEAK**! : NEW STATE = 3

## Step 5: Here's All of the Code

#include "C4.h" // Sample Frequency in Hz const float sample_freq = 22050; int len = sizeof(rawData); int i,k; long sum, sum_old; int thresh = 0; float freq_per = 0; byte pd_state = 0; void <b>setup</b>() { <b>Serial</b>.begin(115200); sum = 0; pd_state = 0; int period = 0; // Autocorrelation for(i=0; i < len; i++) { sum_old = sum; sum = 0; for(k=0; k < len-i; k++) sum += (rawData[k]-128)*(rawData[k+i]-128)/256; // Peak Detect State Machine if (pd_state == 2 && (sum-sum_old) <=0) { period = i; pd_state = 3; } if (pd_state == 1 && (sum > thresh) && (sum-sum_old) > 0) pd_state = 2; if (!i) { thresh = sum * 0.5; pd_state = 1; } } // Frequency identified in Hz freq_per = sample_freq/period; <b>Serial</b>.println(freq_per); } void <b>loop</b>() { // put your main code here, to run repeatedly: }

## Step 6: Files

The C4.h File contains the buffer we're analysing.

You can fill that by reading a Block of Data from the ADC in the Arduino. Or you can generate the data from a program such as Audacity.

Here's a link to the Files including several .h files containing Audio samples you can try out.

The samples were taken from the extensive collection captured by the University of Iowa Electronic Music Studios.

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