Sampling and reconstruction are two of the most essential and widely used operations in signal-processing systems. In this lesson you will be introduced to the roles of sampling and reconstruction in signal processing and the questions that will be addressed in subsequent lessons. All signals in the physical world, e.g., sound or light intensity, have Read More

## Aliasing and the Sampling Theorem Simplified

In this lesson you will learn why aliasing occurs when sampling a signal. Aliasing is when a continuous-time sinusoid appears as a discrete-time sinusoid with multiple frequencies. The sampling theorem establishes conditions that prevent aliasing so that a continuous-time signal can be uniquely reconstructed from its samples. The sampling theorem is very important in signal Read More

## Fourier Transform Interpretation of Sampling

In the Fourier Transform Interpretation of Sampling lesson you will learn how the Fourier transform of the sampled signal depends on the Fourier transform of the original continuous-time signal. This relationship provides the basis for understanding the sampling theorem, how to reconstruct a continuous-time signal from samples, and how aliasing can distort the frequency content of Read More

## Reconstruction and the Sampling Theorem

In this lesson you will learn when a continuous-time signal can be reconstructed from its samples and how to do the reconstruction. These insights are easy to obtain using the frequency domain representation for sampling that is derived in the preceding lesson. The condition for unique reconstruction is that the sampling theorem be satisfied. The Read More

## Reconstruction and the Sampling Theorem Examples

This lesson presents several examples of sampling to illustrate aliasing and the conditions of the sampling theorem. Prerequisites Reconstruction and the Sampling Theorem

## Two-Dimensional Sampling Theorem

The principles that govern sampling of signals in time are easily extended to sampling of two- and higher-dimensional signals. In this lesson the conditions required to satisfy the sampling theorem in two dimensions. Sampling is with respect to space in this case. You will learn how the highest spatial frequency of interest determines the required spacing between Read More

## Equivalent Analog Filtering

Analog filters, composed of electrical components such as resistors, capacitors and op amps, have been used historically to filter continuous-time signals. In this lesson you will learn how a signal-processing system consisting of sampling, discrete-time filtering, and reconstruction is equivalent to an analog filter. Sampling, discrete-time filtering, and reconstruction are often used in place of Read More

## Practical Sampling: Anti-Aliasing Filters

Real physical signals rarely satisfy the conditions of the sampling theorem because often they are not band limited and noise is present. This lesson introduces you to a technique that is used in every practical sampling system. An analog or continuous-time low pass filter is applied to the signal before sampling. This insures the signal Read More

## Practical Reconstruction: The Zero-Order Hold

In practice reconstruction of a continuous-time signal from samples is normally performed using a device called a zero-order hold. The zero-order hold puts out a voltage proportional to the amplitude of the discrete-time signal and holds that value for the duration of the sampling interval. The result is a stair-step approximation to the signal. This Read More

## Practical Digital Filtering and Oversampling

This lesson introduces you to the concept of oversampling – using a significantly higher sampling rate than required by the sampling theorem. Oversampling is motivated by the cost and difficulty of building high-performance analog anti-aliasing and anti-imaging filters. You will also learn how downsampling and upsampling are incorporated into a signal processing system to minimize Read More

## Oversampling Example

This lesson demonstrates how oversampling can improve the quality of sampling and reconstruction. It assumes the anti-aliasing and anti-imaging filters are fixed as a simple, second-order analog filter. Oversampling by a factor of eight reduces the gain of any aliased components by a factor of 76. A similar reduction in the gain of the image Read More

## Downsampling: Reducing the Sampling Rate

In this lesson you will learn how downsampling is implemented to reduce the sampling rate of a signal. You will also learn the effect of downsampling on the discrete-time Fourier transform of the signal. Downsampling is an important method for balancing cost in practical signal processing systems. Prerequisites Practical Digital Filtering and Oversampling

## Upsampling: Increasing the Sampling Rate

Upsampling is the process of discrete-time interpolation. In this lesson you will learn how usampling is implemented to increase the sampling rate of a signal. You will also learn the effect of upsampling on the discrete-time Fourier transform of the signal. Upsampling is an important tool for balancing cost in practical signal processing systems. Prerequisites Read More

## Analog to Digital Conversion: Quantization and Coding

This lesson introduces you to the steps that are involved in converting a signal from analog, or continuous time and continuous amplitude, format to a format that can be manipulated with a digital computer. Digital computers represent numbers using a finite number of bits. This implies that only a finite number of signal levels are Read More

## Analysis of Quantization Error

Quantization introduces error between the original signal and the quantized version. This error is viewed as a noise that contaminates the signal. In this lesson you will learn how quantization noise is usually modeled. You will also learn how the signal-to-quantization-noise ratio depends on the number of bits used to represent the signal. Prerequisites Analog Read More