Signals and Systems¶
Oppenheim, A. V., Willsky, A. S., & Nawab, S. H. (1996). Signals and Systems. Prentice Hall.
Cited by¶
5 citations across 5 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Convolution
- A position-indexed input. There is a signal — a function over time, space, an index, or a probability density — whose values are arrayed over positions a kernel can be slid across. A fixed kernel. A single weight pattern (impulse response, influence function, susceptibility, synaptic weight) specifies how much each input position contributes to a nearby output position; its interpretation is the only thing that changes across substrates. A local mixing operation. Each output value is the flipped, shifted, weighted sum of input values under the kernel — a localized mixture computed identically at every position. Translation invariance. The same kernel applies at every position; convolution is the unique linear operation commuting with translation, so any analysis demanding shift-invariance lands here by necessity.
This sourceDefines the continuous and discrete convolution, the flip-and-slide operation, LTI impulse response, translation invariance, the convolution theorem, and the operation's commutativity/associativity/identity algebra.
- A position-indexed input. There is a signal — a function over time, space, an index, or a probability density — whose values are arrayed over positions a kernel can be slid across. A fixed kernel. A single weight pattern (impulse response, influence function, susceptibility, synaptic weight) specifies how much each input position contributes to a nearby output position; its interpretation is the only thing that changes across substrates. A local mixing operation. Each output value is the flipped, shifted, weighted sum of input values under the kernel — a localized mixture computed identically at every position. Translation invariance. The same kernel applies at every position; convolution is the unique linear operation commuting with translation, so any analysis demanding shift-invariance lands here by necessity.
- Harmonic Distortion
- Harmonic distortion is the structural pattern in which a signal passed through a nonlinear transfer function emerges carrying new frequency components — harmonics at integer multiples of the input frequencies, and intermodulation products at their sums and differences — that were not present in the input at all.
This sourceStandard text: linear systems cannot create new frequencies, while a nonlinear (memoryless) map raises a signal to powers, generating harmonics at integer multiples and intermodulation products at sums and differences.
- Harmonic distortion is the structural pattern in which a signal passed through a nonlinear transfer function emerges carrying new frequency components — harmonics at integer multiples of the input frequencies, and intermodulation products at their sums and differences — that were not present in the input at all.
- Reference Cadence Exceeds Tracking Bandwidth
- The error-spectrum diagnostic is exact and measurable: take the Fourier transform of the tracking error and observe that it is concentrated above 10 Hz — that spectral signature is the proof that the failure is bandwidth mismatch, not a sluggish or mis-tuned controller, because below 10 Hz the same loop tracks faithfully.
This sourceStandard treatment of frequency-domain analysis, spectra of error signals, and aliasing under undersampling.
- The error-spectrum diagnostic is exact and measurable: take the Fourier transform of the tracking error and observe that it is concentrated above 10 Hz — that spectral signature is the proof that the failure is bandwidth mismatch, not a sluggish or mis-tuned controller, because below 10 Hz the same loop tracks faithfully.
- Vector Space
- Signal processing: signals as elements of \(L^2\); Fourier analysis as basis decomposition in a function space.
This sourceSignals as elements of a function space and Fourier analysis as basis decomposition.
- Signal processing: signals as elements of \(L^2\); Fourier analysis as basis decomposition in a function space.
Domain-specific¶
- Harmonic Spectrum
- Square wave. An ideal symmetric square wave contains odd harmonics \(f_0,3f_0,5f_0,\ldots\) with amplitudes decreasing in inverse proportion to harmonic number, the \(n\)th harmonic scaling as \(1/n\)
This sourceSupplies the Fourier-series derivation showing a symmetric square wave's odd harmonics scale in amplitude as 1/n.
- Square wave. An ideal symmetric square wave contains odd harmonics \(f_0,3f_0,5f_0,\ldots\) with amplitudes decreasing in inverse proportion to harmonic number, the \(n\)th harmonic scaling as \(1/n\)
Verification¶
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Links previously used in the corpus¶
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- https://www.pearson.com/en-us/subject-catalog/p/signals-and-systems/P200000003155 ×1
- https://www.pearson.com/en-us/subject-catalog/p/signals-and-systems/P200000003155/9780138147570 ×1
- https://www.pearson.com/en-us/subject-catalog/p/signals-and-systems/P200000003322 ×1
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