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**Unformatted text preview: **Convolution and Linear Systems Convolution and Linear Systems CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science Convolution and Linear Systems Introduction Analyzing Systems Goal: analyze a device that turns one signal into another. Notation: f ( t ) g ( t ) f ( t ) is the input signal g ( t ) is the output signal (Well write this for 1-dimensional signals, but all of the theory applies to higher-dimensional signals as well.) Convolution and Linear Systems Introduction Linearity (Revisited) An operation T is linear if and only if T ( ax + by ) = aT ( x ) + bT ( y ) Implications: T ( ax ) = aT ( x ) T ( x + y ) = T ( x ) + T ( y ) Convolution and Linear Systems Introduction Multiplying an Input to a Linear Operation If the operation is linear: f ( t ) g ( t ) a f ( t ) a g ( t ) Applying a linear operation to an signal multiplied by a constant is the same as applying the operation and then multiplying by that constant. This is called the scaling property of linear operations. Convolution and Linear Systems Introduction Adding Inputs to a Linear Transformation If the operation is linear: f 1 ( t ) g 1 ( t ) f 2 ( t ) g 2 ( t ) f 1 ( t ) + f 2 ( t ) g 1 ( t ) + g 2 ( t ) Applying a linear operation to the sum of two signals is the same as applying it to each separately and adding the results. This is called the superposition property of linear operations. Convolution and Linear Systems Introduction Shift Invariance Shift invariance means an operation is invariant to translation. Implication: If you shift the input, you get the same (shifted) output. In other words, f ( t ) g ( t ) implies f ( t + T ) g ( t + T ) Convolution and Linear Systems Introduction Systems Linearity and shift invariance are nice properties for a signal-processing operation to have: input devices output devices processing A transformation that is both linear and shift invariant is called a system ....

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