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# review - ECE-600 Phil Schniter Introduction Much of modern...

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ECE-600 Phil Schniter September 21, 2010 Introduction: Much of modern engineering is concerned with signals and systems. Roughly, a signal is an information-containing waveform (e.g., audio signal, image, digital video stream) and a system converts one waveform to another (e.g., low pass filter, mp3 encoder, a simple delay). Many real-world signals live in a continuous domain (e.g., continuous-time), while much of modern signal processing is done via computation in a discrete domain. In this course, we develop a solid understanding of signals/systems, especially the relationship between their continuous and discrete representations. The frequency domain will be of particular importance here. This course is about understanding/applying concepts , not memorizing formulas, and mathematics is the language of these concepts. Thus, a major goal of this course is making you comfortable with the mathematics of signals and systems. 1

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ECE-600 Phil Schniter September 21, 2010 Outline: 1. Review of signals/systems background (a) Systems: linear, time-invariant, causal, stable (b) Continuous-time transforms: FS, CTFT, Laplace (c) Discrete-time transforms: DTFT, Z-transform 2. Sampling and reconstruction (a) sampling, aliasing, reconstruction (sinc & practical) (b) upsampling, downsampling, and rate conversion 3. Processing of finite-length signals (a) DFT, circ conv, windowing, spectral analysis (b) matrix/vector formulations (c) FFT, fast convolution, overlap/save 4. Design of discrete-time filters (a) goals: magnitude, group delay, joint (b) FIR designs: window, minimax, least-squares (c) IIR designs: bilinear transform, Prony’s method 2
ECE-600 Phil Schniter September 21, 2010 Signals and systems: Continuous-time: x ( t ) H c y ( t ) signals: { x ( t ) } t and { y ( t ) } t system: H c Discrete-time: x [ n ] H y [ n ] signals: { x [ n ] } n and { y [ n ] } n system: H Uniform sampling: We say that x [ n ] is a T -sampled version of x ( t ) when x [ n ] = x ( nT ) for every integer n . Here, T denotes the sampling interval in seconds. Key question: What “information” does the discrete-time sequence { x [ n ] } n = −∞ contain relative to the continuous-time waveform { x ( t ) } t ( −∞ , ) ? 3

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ECE-600 Phil Schniter September 21, 2010 A few important system properties: For the statements below, assume that y [ n ] = H{ x [ n ] } . 1. Linear : H{ αx [ n ] + βw [ n ] } = α H{ x [ n ] } + β H{ w [ n ] } . 2. Time-invariant : For any shift d , H{ x [ n d ] } = y [ n d ] . 3. Causal : At any time m , the output value y [ m ] does not depend on the future input values { x [ n ] } n>m . 4. BIBO Stable : If input x [ n ] is bounded, then output y [ n ] is bounded. Recall that “bounded x [ n ] ” means that there exists some finite value M x such that | x [ n ] | < M x for all n . Note : These properties can be directly re-stated for continuous-time systems as well. 4
ECE-600 Phil Schniter September 21, 2010 Some system examples: 1. Squaring: y ( t ) = | x ( t ) | 2 linear? time-invariant? causal? stable?

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review - ECE-600 Phil Schniter Introduction Much of modern...

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