Linear time-invariant (LTI) systems are the most widely used systems in signal processing. In this lesson you will develop a deeper understanding for the role of the impulse response of LTI systems. You will learn how linearity and time invariance result in the convolution sum for expressing the output of an LTI system in terms […]

## Linear Phase FIR Filters

A significant benefit of FIR filters is that they can be guaranteed to have linear phase and not to introduce phase distortion. In this lesson you will learn the constraints place on the filter coefficients and frequency response when we require linear phase. For example, you will learn why certain orders cannot be used to […]

## 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 […]

## Fast Fourier Transform (FFT) Algorithm

In this lesson you will learn the principles at the core of the decimation-in-time fast Fourier transform algorithm. The (re)discovery of the fast Fourier transform algorithm by Cooley and Tukey in 1965 was perhaps the most significant event in the history of signal processing. There is evidence that Gauss first developed a fast Fourier transform-type […]

## Jupyter Notebook: Explore Image Filtering

In this activity you will learn how simple image filters are used to modify the character of an image. A simple image filter replaces the value of each pixel with a weighted sum of the values of the neighboring pixels. The weights determine the nature of the filtering and can be chosen to accentuate edges […]

## Building Signals with Blocks: Basis Expansions

The notion of building complex signals using elementary signals – metaphorical “blocks” – is central to many signal processing tools, such as Fourier transforms, wavelet transforms, and principle component analysis. The details of these different tools vary and can appear complex. This lesson presents a unified, big picture view of this topic that will help […]