# [Week 2] Signals and Systems

Things are getting more exciting!

The week is divided to three segments. The first segment introduced some 2D and 3D discrete signals such as Images and videos, mentioning the meaning of Unit Impulse and Unit Step discrete signals, also the segment shows the separation availability between multiple numbers of signals.

In the second segment. It introduced the 2-dimenational complex exponential signal and how to build blocks of signals that have the same frequencies.

In the third segment. It was about 2D convolution examples in which some filters are used to manipulate the original image to produce an output image.

2D and 3D Discrete Signals

• 2D discrete signals are signals that depend on 2 variables, each one of them has its own minimum and maximum values.
• Example: Images
• 2D Image consists of pixels, each pixel has its x-y coordinate in the image
• The pixel holds 3 combined values which represent the RED, GREEN and BLUE values
• If the image is Gray-scaled, this means that RED == GREEN == BLUE
• 3D discrete signals have an additional variable
• Videos could be considered 3D discrete signals because the dimensions are: x, y, z, where is z is the frame number in the video, and x, y are the dimension of the frame

Discrete Unit Impulse

• Impulse signal is a function which value is zero everywhere except at zero
• So, the discrete unit impulse of 2 signals is 1 if the values of the first and second signals are zero
• The discrete unit impulse of 2 signals n1, n2 is
• The signals are called “Separable” if

Discrete Unit Step

• The discrete unit step of 2 signals is 1 if the values of the first and second signals are greater or equal zero
• The discrete unit step of 2 signals, n1, n2 is

Complex Exponential Signals

• The complex exponential of 2 signals are defined as
• Where  are the periodicity of signals respectively
• According to Euler formula,
• Where  is a rational multiple of

2D Systems

• Systems that takes  as an input, and using some operations or filters T[input], it produces
• There are 3 kinds of systems
• Systems are : Linear Systems, Spatially Invariant Systems and Linear and Spatially Invariant Systems

Linear System

• Given the input  where   is a weight
• The system is called “Linear System” if the output of the input is

Spatially Invariant Systems

• Given
• The system is called “Spatially Invariant” if

Linear and Spatially Invariant Systems

• Depends on the impulse response of the sight signal, the sight signal could be obtained from devices such as Camera, Telescope
• Let  be the impulse response of sight signal
• Now, for any given input  to the impulse response, the output is  , where ** is 2D discrete convolution

Examples for 2D convolution

• Noise Reduction
• Edge Detection
• Sharpen
• Blur

References