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SIG:010117The scaled reversed triangle x(2-3t) lives on [1/3,1], has energy 2/9, and is not an LTI test

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Sas Triangle Scale Reverse SupportSas Time Scale Energy Jacobian

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Signal spaces and LTI systemsSignals and Systems · Mixed review

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SIG:010118The everlasting 3 cos(2 pi t) has power 9/2; its [-1,1] window has energy 9; the two-tone sequence has period 12 and power 1

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Sas Energy Power Signal ClassSas Discrete Period Average Power

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Signal spaces and LTI systemsSignals and Systems · Mixed review

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SIG:010219Linear convolution of [1,-2,1] with [1,1,1] is [1,-1,0,-1,1] and deconvolution recovers the input

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Sas Discrete Linear Convolution SumSas Fir Generating Function Deconvolution

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Continuous and discrete convolutionSignals and Systems · Mixed review

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SIG:010220The causal convolution e^{-2t}u * e^{-5t}u peaks at ln(5/2)/3; the equal-pole limit is t e^{-2t}u

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Sas Causal Exponential Overlap IntegralSas Repeated Pole Convolution Limit

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Continuous and discrete convolutionSignals and Systems · Mixed review

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SIG:010221Sifting h=delta(t)+2 delta(t-1)-delta(t+2) against e^{-t}u produces breakpoints -2,0,1 and integral 2

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Sas Impulse Train Distributional SiftingSas Step Breakpoint Piecewise Audit

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Continuous and discrete convolutionSignals and Systems · Mixed review

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SIG:010322Time weighting differentiates the spectrum and changes its energy.

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Time Weight FT DerivativeParseval Energy Time Weight

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Fourier series and transformsSignals and Systems · Mixed review

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SIG:010323Modulation shifts two spectra, but overlapping lobes keep a cross term.

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Cosine Modulation ShiftOverlapping Lobe Cross Energy

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Fourier series and transformsSignals and Systems · Mixed review

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SIG:010424The unilateral z-transform must carry the sample before zero.

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Unilateral Z Delay PropertyInitial Rest Recurrence Solve

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Laplace and z transformsSignals and Systems · Mixed review

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SIG:010425A repeated pole creates the polynomial factor, not a second exponential.

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Repeated Pole N Plus OneGeometric Polynomial Energy Sum

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Laplace and z transformsSignals and Systems · Mixed review

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SIG:010526Sampling can be alias-free while the hold still droops and delays the waveform.

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Zoh Frequency ResponseSinc Droop Linear Phase

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Sampling, aliasing, and reconstructionSignals and Systems · Mixed review

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SIG:010527Alternating clock jitter creates a deterministic image spur, not random noise

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Deterministic Alternating Jitter ExpansionDiscrete Image Spur Amplitude

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Sampling, aliasing, and reconstructionSignals and Systems · Mixed review

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SIG:010628Stable eigenvalues can amplify the state before every mode decays

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Matrix Exponential Upper TriangularNonnormal Transient Peak

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State-space systems, causality, and stabilitySignals and Systems · Mixed review

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SIG:010629Observability is quantitative: the Gramian prices every initial direction

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Observability Gramian IntegralOutput Energy Quadratic Form

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State-space systems, causality, and stabilitySignals and Systems · Mixed review

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SIG:010630One input-output energy ratio is not the induced L2 gain

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Induced L2 H Infinity NormOne Input Energy Ratio Gap

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State-space systems, causality, and stabilitySignals and Systems · Mixed review

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SIG:010631The finite-horizon Gramian finds the least-energy input pair exactly

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Finite Horizon Reachability GramianMinimum Energy Steering Input

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State-space systems, causality, and stabilitySignals and Systems · Mixed review

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926011Let n\in\mathbb{Z}.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926012A real continuous-time signal x(t) is defined by The even and odd parts are defined pointwise for every real t by D…

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926013A real discrete-time sequence x[n] is zero except at The even and odd parts are defined for every integer n by Comp…

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926014Let x and y be real-valued continuous-time signals defined on \mathbb{R}.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926015The energy of a continuous-time signal x(t) is defined by provided the integral exists in [0,\infty].

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926016The average power of a continuous-time signal x(t) is when the limit exists in [0,\infty].

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926017The unit step is defined by u(t)=0 for t<0 and u(t)=1 for t\ge 0.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926018Let t\in\mathbb{R} be measured in seconds, and let amplitudes be in volts.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926019The energy and average power of a discrete-time sequence are when the sum and limit exist in [0,\infty].

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926020The energy and average power of a discrete-time sequence are when the sum and limit exist in [0,\infty].

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926021The discrete-time unit impulse and unit step are The continuous-time unit impulse \delta(t) satisfies \delta(t)=0 f…

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926022The unit step satisfies u(t)=0 for t<0 and u(t)=1 for t\ge 0.

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926023Use the continuous-time energy and average power Amplitudes are in volts and t is in seconds.

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926024For each signal below, decide independently: (i) periodic or not periodic, and if periodic the fundamental period;…

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926025Let x be a real continuous-time signal with finite energy Let a\in\mathbb{R} and b\in\mathbb{R}\setminus\{0\}, and…

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926026The energy and average power of a discrete-time sequence are when the sum and limit exist in [0,\infty].

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926027Energy of a possibly complex signal uses the modulus: and the corresponding average powers are the two-sided limits…

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926028A real continuous-time signal x(t) is periodic with period T=10\,\mathrm{s}, so x(t+10)=x(t) for every real t.

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926029Continuous-time time scaling x(t/2) is defined for every real t whenever x is defined on \mathbb{R}.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926030Each part constructs or uniquely determines a signal from stated symmetry, support, periodicity, energy, or power c…

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926031Throughout this problem, continuous-time systems map real-valued functions of t\in\mathbb{R} to real-valued functio…

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926032Let x be a real-valued, locally integrable input voltage, measured in volts, with time t in seconds.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926033Let inputs and outputs be real voltages as functions of t\in\mathbb{R} with t in seconds.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926034Time t is measured in seconds and the signals x and y are voltages in volts.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926035Let u(t)=1 for t\ge 0 and u(t)=0 for t<0, and let \delta denote the Dirac delta.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Problem library

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Topic choices for Signals and Systems are shortened. Search still covers every problem.

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SIG:010117The scaled reversed triangle x(2-3t) lives on [1/3,1], has energy 2/9, and is not an LTI test

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Sas Triangle Scale Reverse SupportSas Time Scale Energy Jacobian

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Signal spaces and LTI systemsSignals and Systems · Mixed review

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SIG:010118The everlasting 3 cos(2 pi t) has power 9/2; its [-1,1] window has energy 9; the two-tone sequence has period 12 and power 1

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Sas Energy Power Signal ClassSas Discrete Period Average Power

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Signal spaces and LTI systemsSignals and Systems · Mixed review

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SIG:010219Linear convolution of [1,-2,1] with [1,1,1] is [1,-1,0,-1,1] and deconvolution recovers the input

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Sas Discrete Linear Convolution SumSas Fir Generating Function Deconvolution

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Continuous and discrete convolutionSignals and Systems · Mixed review

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SIG:010220The causal convolution e^{-2t}u * e^{-5t}u peaks at ln(5/2)/3; the equal-pole limit is t e^{-2t}u

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Sas Causal Exponential Overlap IntegralSas Repeated Pole Convolution Limit

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Continuous and discrete convolutionSignals and Systems · Mixed review

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SIG:010221Sifting h=delta(t)+2 delta(t-1)-delta(t+2) against e^{-t}u produces breakpoints -2,0,1 and integral 2

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Sas Impulse Train Distributional SiftingSas Step Breakpoint Piecewise Audit

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Continuous and discrete convolutionSignals and Systems · Mixed review

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SIG:010322Time weighting differentiates the spectrum and changes its energy.

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Time Weight FT DerivativeParseval Energy Time Weight

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Fourier series and transformsSignals and Systems · Mixed review

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SIG:010323Modulation shifts two spectra, but overlapping lobes keep a cross term.

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Cosine Modulation ShiftOverlapping Lobe Cross Energy

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Fourier series and transformsSignals and Systems · Mixed review

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SIG:010424The unilateral z-transform must carry the sample before zero.

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Unilateral Z Delay PropertyInitial Rest Recurrence Solve

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Laplace and z transformsSignals and Systems · Mixed review

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SIG:010425A repeated pole creates the polynomial factor, not a second exponential.

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Repeated Pole N Plus OneGeometric Polynomial Energy Sum

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Laplace and z transformsSignals and Systems · Mixed review

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SIG:010526Sampling can be alias-free while the hold still droops and delays the waveform.

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Zoh Frequency ResponseSinc Droop Linear Phase

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Sampling, aliasing, and reconstructionSignals and Systems · Mixed review

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SIG:010527Alternating clock jitter creates a deterministic image spur, not random noise

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Deterministic Alternating Jitter ExpansionDiscrete Image Spur Amplitude

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Sampling, aliasing, and reconstructionSignals and Systems · Mixed review

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SIG:010628Stable eigenvalues can amplify the state before every mode decays

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Matrix Exponential Upper TriangularNonnormal Transient Peak

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State-space systems, causality, and stabilitySignals and Systems · Mixed review

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SIG:010629Observability is quantitative: the Gramian prices every initial direction

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Observability Gramian IntegralOutput Energy Quadratic Form

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State-space systems, causality, and stabilitySignals and Systems · Mixed review

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SIG:010630One input-output energy ratio is not the induced L2 gain

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Induced L2 H Infinity NormOne Input Energy Ratio Gap

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State-space systems, causality, and stabilitySignals and Systems · Mixed review

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SIG:010631The finite-horizon Gramian finds the least-energy input pair exactly

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Finite Horizon Reachability GramianMinimum Energy Steering Input

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State-space systems, causality, and stabilitySignals and Systems · Mixed review

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926011Let n\in\mathbb{Z}.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926012A real continuous-time signal x(t) is defined by The even and odd parts are defined pointwise for every real t by D…

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926013A real discrete-time sequence x[n] is zero except at The even and odd parts are defined for every integer n by Comp…

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926014Let x and y be real-valued continuous-time signals defined on \mathbb{R}.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926015The energy of a continuous-time signal x(t) is defined by provided the integral exists in [0,\infty].

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926016The average power of a continuous-time signal x(t) is when the limit exists in [0,\infty].

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926017The unit step is defined by u(t)=0 for t<0 and u(t)=1 for t\ge 0.

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926018Let t\in\mathbb{R} be measured in seconds, and let amplitudes be in volts.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926019The energy and average power of a discrete-time sequence are when the sum and limit exist in [0,\infty].

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926020The energy and average power of a discrete-time sequence are when the sum and limit exist in [0,\infty].

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926021The discrete-time unit impulse and unit step are The continuous-time unit impulse \delta(t) satisfies \delta(t)=0 f…

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926022The unit step satisfies u(t)=0 for t<0 and u(t)=1 for t\ge 0.

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926023Use the continuous-time energy and average power Amplitudes are in volts and t is in seconds.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926024For each signal below, decide independently: (i) periodic or not periodic, and if periodic the fundamental period;…

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926025Let x be a real continuous-time signal with finite energy Let a\in\mathbb{R} and b\in\mathbb{R}\setminus\{0\}, and…

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926026The energy and average power of a discrete-time sequence are when the sum and limit exist in [0,\infty].

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926027Energy of a possibly complex signal uses the modulus: and the corresponding average powers are the two-sided limits…

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926028A real continuous-time signal x(t) is periodic with period T=10\,\mathrm{s}, so x(t+10)=x(t) for every real t.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926029Continuous-time time scaling x(t/2) is defined for every real t whenever x is defined on \mathbb{R}.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926030Each part constructs or uniquely determines a signal from stated symmetry, support, periodicity, energy, or power c…

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926031Throughout this problem, continuous-time systems map real-valued functions of t\in\mathbb{R} to real-valued functio…

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926032Let x be a real-valued, locally integrable input voltage, measured in volts, with time t in seconds.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926033Let inputs and outputs be real voltages as functions of t\in\mathbb{R} with t in seconds.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926034Time t is measured in seconds and the signals x and y are voltages in volts.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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SIG:9926035Let u(t)=1 for t\ge 0 and u(t)=0 for t<0, and let \delta denote the Dirac delta.

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Signals LtiSignals And Systems Problem Solving

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Signal spaces and LTI systemsSignals and Systems · Calculation

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Problem results

SIG:9926001A continuous-time voltage signal x(t), with t in seconds, is defined for every real t by Evaluate x(-2), x(-1.5), x…

SIG:9926002A discrete-time sequence x[n] is defined for every integer n by List every pair (n,x[n]) for which x[n]\ne 0.

SIG:9926003A continuous-time signal w(t), with t in seconds, is given by Define the delayed signal w_d(t)=w(t-2) and the advan…

SIG:9926004A continuous-time signal w(t), with t in seconds, is given by Define the time-reversed signal v(t)=w(-t) for every…

SIG:9926005A continuous-time rectangular pulse g(t), with t in seconds, is defined by Define g_c(t)=g(2t), g_e(t)=g(t/3), and…

SIG:9926006A continuous-time signal p(t), with t in seconds, is defined by Define q(t)=p(6-2t) for every real t.

SIG:9926007A discrete-time sequence x[n] is zero except at the following samples: (the listed value x[-2]=0 is included only f…

SIG:9926008Throughout this problem, t\in\mathbb{R} is measured in seconds.

SIG:9926009Throughout this problem, n\in\mathbb{Z}.

SIG:9926010Let t\in\mathbb{R} be measured in seconds.

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Problem results

SIG:9926001A continuous-time voltage signal x(t), with t in seconds, is defined for every real t by Evaluate x(-2), x(-1.5), x…

SIG:9926002A discrete-time sequence x[n] is defined for every integer n by List every pair (n,x[n]) for which x[n]\ne 0.

SIG:9926003A continuous-time signal w(t), with t in seconds, is given by Define the delayed signal w_d(t)=w(t-2) and the advan…

SIG:9926004A continuous-time signal w(t), with t in seconds, is given by Define the time-reversed signal v(t)=w(-t) for every…

SIG:9926005A continuous-time rectangular pulse g(t), with t in seconds, is defined by Define g_c(t)=g(2t), g_e(t)=g(t/3), and…

SIG:9926006A continuous-time signal p(t), with t in seconds, is defined by Define q(t)=p(6-2t) for every real t.

SIG:9926007A discrete-time sequence x[n] is zero except at the following samples: (the listed value x[-2]=0 is included only f…

SIG:9926008Throughout this problem, t\in\mathbb{R} is measured in seconds.

SIG:9926009Throughout this problem, n\in\mathbb{Z}.

SIG:9926010Let t\in\mathbb{R} be measured in seconds.