EECE 301 Review - EECE301 Review Concepts, Big Ideas, &...

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1 EECE301 Review Concepts, Big Ideas, & Connections
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2 Signals… Time-Domain Functions • Some Common Signals – Sinusoids… • real-valued… the things we see in the real world • complex-valued… the math we use in Fourier Methods –Un i t S tep u ( t ) • Good test signal for system response • Good for “building” signals by turning on/off others (2 ) ) cos(2 ) 2 oo jf t t o A Af t e e πφ +−+ ⎡⎤ += + ⎣⎦ 1, 0 () 0, 0 t ut t = <
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3 Signals… Time-Domain Functions • Some Common Signals – Delta Function or Impulse δ ( t ) – Exponentials •La rge λ = Fast Decay Small λ = Slow Decay – Decaying Oscillation () ( ) ( ) tx t d t x δ ττ −∞ −= t eu t λ cos(2 ) t o eA f t π φ +
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4 Signals… Time-Domain Functions • Periodic Signals – Sinusoids… are periodic – All other periodic Signals are made from sinusoids • This is the main idea of Fourier Series – Fundamental Period: T = Time of One Cycle – Fundamental Frequency: f o = 1/ T ω o = 2 π / T • Modulated Sinusoid () ()cos (2 ) o x t f t f t π =
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5 Signals… Time-Domain Functions • Time Shifting & Reversal of Signals – Good for “Building” signals – Key part of Convolution
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6 Signals… Time-Domain Functions • DT Signals… come from ADC – Sampling Rate F s = 1/ T s F s 2 f max x [ n ] = x ( nT s ) Sensor Analog Electronics ADC DSP Computer Physical C-T Signal Electrical C-T Signal Electrical C-T System Electrical C-T Signal Electrical D-T Signal Electrical D-T System Electrical D-T Signal
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7 Systems… Signal In, Signal Out • We model circuits and other things as a box – Some wiggling thing goes in – Some wiggling thing comes out System View Math Model of System Math Function for Input x ( t ) y ( t ) Math Function for Output Many Different Math Models • Differential Eq. or Difference Eq. • Convolution with Impulse Response • Frequency Response • Transfer Function Time Domain Freq. Domain
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8 Systems… Signal In, Signal Out • What kind of system did we mostly study? – Linear Time-Invariant (LTI) – Linear Systems satisfy… – Time-Invariant Systems satisfy…. – We considered both Causal & Non-Causal Systems • But ONLY causal systems can exist in real world • Causal Systems satisfy….
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9 Systems… Signal In, Signal Out • Output of an LTI system consists of two parts – Zero-Input Solution – Zero-State Solution Recall: Two parts to the solution (i) one part due to ICs with zero-input (“zero-input response”) (ii) one part due to input with zero ICs (“zero-state response”) “Homogeneous Solution” Then : y ( t ) = y ZI ( t ) + y ZS ( t ) Same for DT Systems… Can find using : • Char. Roots & Modes • Part of LT or ZT “Complete” Soln Can find using : • Convolution • Frequency Response • Transfer Function
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10 Differential equations like this are LTI - coefficients ( a ’s & b ’s) are constants TI - No nonlinear terms Linear ) ( ) ( ...
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EECE 301 Review - EECE301 Review Concepts, Big Ideas, &amp;...

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