systems-harmonics.slides.printing.6

# systems-harmonics.slides.printing.6 - When a system is...

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Generalized Harmonic Functions Generalized Harmonic Functions CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science Generalized Harmonic Functions Harmonic Functions What does f ( t ) = e i 2 π ut look like? Imaginary Real i 1 -i -1 Generalized Harmonic Functions Harmonic Functions as Sinusoids Real Part Imaginary Part ± ( e i 2 π ut ) ² ( e i 2 π ut ) cos ( 2 π ut ) sin ( 2 π ut ) So, e i 2 π ut is just a way to write a cosine function (real part) and a sine function (imaginary part) simultaneously—each has frequency u . This is called a generalized harmonic . Generalized Harmonic Functions Harmonics and Systems What happens when we feed a generalized harmonic into a linear, shift-invariant system? f ( t ) = e i 2 π ut f ( t ) H ( u ) f ( t ) where H ( u ) is a complex multiplier.
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Unformatted text preview: When a system is applied to a harmonic signal, the result is the same harmonic signal multiplied by a complex constant that depends on the frequency. Generalized Harmonic Functions Transfer Functions H ( u ) is called the transfer function : how inputs transfer to outputs. Expressing H ( u ) in polar (magnitude-phase) form: H ( u ) = | H ( u ) | e i φ ( H ( u )) Recall that the magnitudes multiply and the phases add: H ( u ) e i 2 π ut = | H ( u ) | e i φ ( u ) e i 2 π ut = | H ( u ) | e i ( 2 π ut + φ ( u )) | H ( u ) | is the Modulation Transfer Function (MTF) φ ( H ( u )) is the Phase Transfer Function (PTF)...
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