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Unformatted text preview: 5 Repeated roots 5.1 Leaky-tank background 64 5.2 Numerical computation 65 5.3 Analyzing the output signal 67 5.4 Deforming the system: The continuity argument 68 5.5 Higher-order cascades 70 After reading this chapter you should be able • to use a continuity argument • to explain the non-geometric output of a mode with a double root. Modes generate persistent outputs. So far our examples generate persis- tent geometric sequences. But a mode from a repeated root, such as from the system functional ( 1 − R /2 ) − 3 , produces outputs that are not geomet- ric sequences. How does root repetition produce this seemingly strange behavior? The analysis depends on the idea that repeated roots are an unlikely, spe- cial situation. If roots scatter randomly on the complex plane, the prob- ability is zero that two roots land exactly on the same place. A generic, decent system does not have repeated roots, and only through special con- trivance does a physical system acquire repeated roots. This fact suggests deforming a repeated-root system into a generic system by slightly moving one root so that the modes of the deformed system produce geometric se- quences. This new system is therefore qualitatively easier to analyze than is the original system, and it can approximate the original system as closely as desired. This continuity argument depends on the idea that the world 64 5.1 Leaky-tank background changes smoothly: that a small change to a system, such as by moving a root a tiny amount, produces only a small change to the system’s output. To generate a double root, we use the RC circuit ( ?? ) or the leaky tank (Section 1.2). Either system alone has one mode. To make a double root, cascade two identical tanks or RC circuits, where the output of the first system is the input to the second system. Exercise 26. When making an RC cascade system analogous to a cascade of two leaky tanks, does the RC cascade need a unity-gain buffer between the two RC cir- cuits? 5.1 Leaky-tank background Let the leaky tank or RC circuit have time constant τ . Let V in be the input signal, which is the flow rate into the tank system or the input voltage in the RC circuit, and let V out be the output signal. Then the differential equation of either system is τ ˙ V out = V in − V out . Convert this equation into a discrete-time system using the forward-Euler approximation ( ?? ). Using a time step T , the difference equation becomes τ V out [ n ] − V out [ n − 1 ] T = V in [ n − 1 ] − V out [ n − 1 ] ....
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- Spring '11
- Binomial Theorem, RC circuit, Impulse response, CASCADE