3 Case Study: Intruder Detection System Using Discrete Fourier Transform (DFT) by Correlation

This section describes the application of a sine wave generator in computing correlation for a DFT-based intruder detection system.

An intruder detection system basically consists of a transmitter which emits a signal at a specific frequency whenever an intruder is found inside a room or an enclosed space.

In most intruder-based systems, the Infrared (IR) signals are used. The receiver receives the signal and detects if any specific frequency component exists. In this detection process, usually a DFT is performed on the received signal and is checked for the presence of the frequency component of interest. When the DFT is implemented using the Correlation method, the sine and cosine waves are required at the frequency of interest.

The sine and cosine waves used in the DFT are called as DFT basis functions. The output of the DFT is a set of numbers that represent amplitudes. The DFT basis functions are a set of sine and cosine waves with unity amplitude. In the frequency domain, if each of the amplitudes is assigned to the sine or cosine waves, the outcome will be a set of sine and cosine waves that can be added to form the time domain signal.

Figure   1 illustrates a typical block diagram of the intruder detection system. An NCO module is configured to produce a square wave of the desired frequency to be detected by the receiver. The output of the NCO is passed through a Band Pass Filter with a suitable frequency band to allow only the frequencies of interest around the center frequency, which is the frequency to be detected. The Band Pass Filter must have a high Q factor to get a better and sharper cutoff around the corner frequencies. Therefore, the output of the Band Pass Filter will be a sine wave at the fundamental frequency.

Figure 3-1. Block Diagram of Intruder Detection System
The ADC inside the PIC® MCU has two inputs: the sine wave output of the Band Pass Filter, and the signal sent from the TX and received at the RX. This signal must be correlated to determine if the frequency of interest (fdetect) exists. After the digital samples from the ADC are obtained, the sine samples are then passed through an orthogonal signal generation process to produce a cosine output. This orthogonal signal generation is computed in the firmware and some of the transforms or filters are described below:
  • Hilbert Transforms: This transform is used to produce output signals which are 90° out of phase with respect to the input signal (i.e, orthogonal to each other). Therefore, if a sine wave is applied at the input, the result will be a cosine wave at the output with no attenuation.
  • Low Pass Filter: If a low pass filter of first order is designed such that the user operates it in the stop band (i.e, beyond the cutoff frequency), and an input sine wave is provided to this filter, then the resulting output will be an attenuated signal of the same wave shape as the input, but shifted in phase by 90°. This signal can then be amplified in the firmware by multiplying with an appropriate gain to get the output amplitude to be same as the input amplitude. Therefore, the resulting waveform will be similar to the input waveform in shape (sine wave), frequency and amplitude, but shifted in phase by 90° (cosine wave).
  • Shifting the Sampled Array: If the sine wave samples (elements in the array) are shifted appropriately such that the output samples are shifted by 90°, the resultant waveform will be a cosine wave
Because the sine wave and cosine waveforms have been generated and are available in digital form, the correlation is performed on the input signal by performing the following computations:
  • Summation of the product of the individual sine wave samples and the input signal samples
  • Summation of the product of the individual cosine wave samples and the input signal samples
Once the output (Y1 and Y2) of the two summations are available, check if the frequencies exist by interpreting the following results:
  • If the frequency to be detected does not exist in the received signal, then the sum of the sine bins and cosine bins will be zero. Otherwise, there will be a finite value.
  • If the sum of the sine bin is finite and the cosine bin is zero, then the signal at the detection frequency exists and the phase shift is zero
  • If the sum of the sine bin is zero and the cosine bin is finite, then the signal at the detection frequency exists and has a phase shift of 90°
  • If the sum in the sine bin and the cosine bin both have finite values, then the signal at the detection frequency exists and the phase shift is finite

The interpreted results are illustrated in Figure   2.

Figure 3-2. Frequency Detection Principle in Intruder Detection System