Information Transmission Chapter 2, repetition OVE EDFORS - - PowerPoint PPT Presentation
1 Information Transmission Chapter 2, repetition OVE EDFORS ELECTRICAL AND INFORMATION TECHNOLOGY 2 Linear, time-invariant (LTI) systems A system is said to be linear if, whenever an input yields an output and an input
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OVE EDFORS ELECTRICAL AND INFORMATION TECHNOLOGY
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A system is said to be linear if, whenever an input yields an output and an input yields an output We also have where are arbitrary real or complex constants.
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Superposition:
is the same as
signals are acting separately.
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When the duration of our pulse approaches 0, the pulse approaches the delta function (also called Dirac's delta function or, the unit impulse)
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The delta function is defined by the property where g(t) is an arbitrary function, continuous at the origin.
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The integral is called convolution and is denoted The output y(t) of a linear, time-invariant system is the convolutional of its input x(t) and impulse response h(t).
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Consider an LTI system with impulse response and input What is the output?
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In school we all learned about complex numbers and in particular about Euler's remarkable formula for the complex exponential Where is the real part is the imaginary part
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is called the frequency function or the transfer function for the LTI system with impulse response h(t).
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The frequency function is in general a complex function of the frequency: where is called the amplitude function and is called the phase function.
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. 2 . 4 . 6 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 x 1
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5 F r e q u e n c y ( H z ) F r e q u e n c y r e s p ( d B )
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. 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5 x 1
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D e l a y ( s ) I m p u l s e r e s p ( d B )
What are the delays? How is the signal affected for different delays? How does it change with time?
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The Fourier transform of the signal x(t) is given by the formula This function is in general complex: where is called the spectrum of x(t) and its phase angle.
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Hence we have a Fourier transform pair
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Since the output y(t) of an LTI system is the convolution of its input x(t) and impulse response h(t) it follows from Property 10 (Convolution in the time domain) that the Fourier transform of its output Y(f) is simply the product of the Fourier transform of its input X(f) and its frequency function H(f), that is,
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Linear system to analyzed described in tme domain Transformed linear system described in frequency domain Soluton or analysis in tme domain Soluton or analysis in frequency domain E.g. convolution The detour may be a lot simpler Multiplication TIME DOMAIN FREQUENCY DOMAIN
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