The applet provides with this www page is an interactive demonstration that introduces the basics of sampling theory.
Quick start.
The signals we use in the real world, like our voices, are called “analog” signals. To process these signals in computers, we need to convert them into a “digital” form. While an analog signal is continuous in both time and amplitude, a digital signal is discrete in both time and amplitude, a digital signal is discrete in both time and amplitude. To convert a signal from continuous time to discrete time, a process called sampling is used. The value of the signal is measured at certain time intervals. Each measurement is called a sample. (The analog signal is also quantized in amplitude, but this process is ignored in this demo).
When the continuous analog signal is sampled at a frequency F, the resulting discrete signal has more frequency components than the analog signal. More precisely, the frequency components of the analog signal are repeated at the sampling rate. That is, in the discrete frequency response, they are seen as their original position, and also centered +/- and +/-2F etc.
How many samples are needed to make sure that we get the information contained in the signal? If the signal contains high frequency components, we need to sample at a higher sampling rate to avoid losing information in the signal. In general, it is necessary to measure at twice the maximum frequency of the signal to obtain full information in the signal. This is called the Nyquist Rate. Sampling theory says that a signal can be accurately reproduced if it si sampled ar a frequency F, where F is greater than twice the maximum frequency in the signal.
What happens if we sample the signal at a frequency lower than the Nyquist Rate? When the signal is converted back to the continuous time signal, it shows a phenomenon called aliasing. Aliasing is the presence of unwanted components in the reconstructed signal. These components were not present when the original signal was sampled. In addition, some of the frequencies in the original signal may be lost in the reconstructed signal. Aliasing occurs because signal frequencies can overlap if the sampling frequency is too low. Frequencies “fold” by half the sampling frequency, which is why this frequency is often referred to as the folding frequency.
Sometimes the highest-frequency components of a signal are simply noise or contain no useful information. To avoid aliasing these frequencies, we can filter out these components before sampling the signal. Because we filter out high-frequency components and let low-frequency components pass trough, we speak of low-pass filtering.
Sampling demo.
The original signal in the following applet consists of three sinusoidal functions, each with a different frequency and amplitude. The example here has the frequencies 28 Hz, 84 Hz and 140 Hz. Use the Filter Controls to filter out the higher frequency components. This filter is an ideal low pass filter, i.e. it keeps exactly all frequencies below the cutoff frequency and completely attenuates all frequencies above the cutoff frequency.
Note that if you leave all components in the original signal and select a low sampling frequency, aliasing will occur. This aliasing will cause the reconstructed signal not to match the original signal. However, you can try to limit the number of aliases by filtering out the higher frequencies in the signal. It is also important to note that if you sample at a rate above the Nyquist rate, further increases in the sampling frequency will not improve the quality of the reconstructed signal. This is due to the ideal low pass filter. In practice, sampling at higher frequencies leads to better reconstructed signals. However, higher sampling frequencies require faster converters and more memory. Therefore, engineers must weigh the pros and cons of each application and be aware of the tradeoffs involved.
The importance of frequency domain plotting for signal analysis cannot be underestimated. The three diagrams on the right side of the demore are all Fournier transform diagrams. It is easy to see the effects of a change in sampling frequency when looking at these transformation plots. As the sampling frequency decreases, the signal separation also decreases. If the sampling frequency falls below the Nyquist rate, the frequencies overlap and cause aliasing.
Instructions for using the system.
The applet is divided into three sections, the Original Signal Panel, the Sampled Digital Signal Panel and the Reconstruted Analog Signal Panel. By selecting the sampling frequencies, you can see the effects of aliasing in the frequency domain plots. By selecting the filter frequency, you can control which signals are left when sampling the analog signal. You can place the original plot over the reconstructed plot if you want to see how different the results are. You can also use the Reset button to reset all values to their original defaults.
Thank you for visiting.