Signals and Systems from Basics to Advance Level

Classify systems as linear or nonlinear by applying the superposition and homogeneity principles, illustrating input-output relations with amplifiers, square roots, exponentials, and time-shift examples.

Analyze time variant and time invariant systems by applying the delayed input vs delayed response test, with examples of shifting, scaling, and integration.

Differentiate static and dynamic systems by memory, where static are memoryless and dynamic rely on past inputs, and define causal versus noncausal behavior by future values.

Classify systems as static or dynamic and causal or noncausal by examining output dependence on present, past, and future inputs, with initial conditions, memory, derivatives, and integrators illustrated.

Explain the stability criterion: a system is stable when a bounded input yields a bounded output, and classify signals as bounded or unbounded, with rectangular, triangular, and unit step examples.

Examine stability of five systems by evaluating whether bounded inputs yield bounded outputs, noting finite versus unbounded responses and cases involving input signals, output signals, and integration effects.

Explore invertible and non-invertible systems by testing whether different inputs yield the same or different outputs, using input-output mappings and scaling examples.

Explore gate previous problems with solutions on signals and systems, analyzing step and impulse responses, convolution, linearity, time invariance, stability, and causality.

Explore GATE previous problems on signals and systems, analyzing causality and stability from impulse and step responses, and solving using convolution and impulse input scenarios.

Solve GATE previous problems set 3 on signals and systems, deriving impulse responses, performing convolutions, and assessing linear time-invariant, causal versus noncausal systems and their stability.

Analyze the input-output relation and delayed responses to classify the system as time-invariant and causal, confirming it relies on present and past inputs, not future ones.

Analyze input-output relations to test linearity using superposition and homogeneity. Examine causality by checking if present output depends on future input, concluding a linear and causal case.

Apply superposition and homogeneity to test linearity of the input-output relation, then use bounded-input bounded-output criteria to assess stability; the example shows a linear but unstable system.

Analyze a system's behavior by applying superposition and homogeneity to verify linearity, then test causality to determine if the system is linear and non-causal, as discussed in IES Part 4.

Assess a discrete-time system from a 2008 paper to decide if it is memoryless or has memory, and test instability by varying inputs and observing behavior at infinity.

Explore a linear time-invariant discrete-time system using unit step and ramp inputs, derive the transfer function, and analyze the ramp response to identify the correct output.

Derive a system's impulse response from its transfer function using the z-transform and inverse z-transform. Analyze the time-domain relation and extract coefficients like 0.2 to form the transfer function.

The discussion explains that a linear system must satisfy both the superposition and homogeneity principles; since only superposition is given, the system may be linear or nonlinear depending on homogeneity.

Analyze a system's input-output relation, verify linearity via the superposition and homogeneity principles, examine time invariance under input scaling, and identify option B as the answer.

Examine time-variant and time-invariant behavior in discrete-time systems, through the IES discussion part 11, using input-output relations and delays to classify systems and select the correct option.

Analyze a discrete-time system with an exponentially decaying impulse response to determine causality and stability. The response is zero for n<0, proving causality, and the sum of (1/2)^n proves stability.

Analyze an impulse response to test causality and stability, using three different functions to show a non-causal system that is stable because the sum of samples is finite.

Explore the relation between step response and impulse response, showing the step response equals the integral of the impulse response and the impulse response is its derivative.

Explore how unit step and RAM function interact with delta function through integration and convolution, revealing singularity functions and the unity step relationship in signals and systems.

Evaluates a system where output is the square of the input, showing nonlinearity, and analyzes linearity, superposition, causality, and time invariance with discrete-time examples.

Examine a system where output equals input multiplied by a constant plus an additive constant, and verify the superposition and homogeneity principles fail, indicating a nonlinear system.

Assess a system’s linearity, homogeneity, and causality by applying the superposition principle and the input-output expression, determining if the system is linear and causal.

Explore the shifting property of the impulse function and its impact on integration and delta function relations, validating expressions and identifying the correct option.

Explore impulse functions and the sampling property to evaluate a discrete-time signal multiplied by delta functions, noting that samples outside minus five to plus five contribute nothing, yielding zero.

Analyze how the impulse response determines a system’s stability and causality, highlighting a right-sided, causal system. The lecture concludes the system is causal and unstable, identifying option B.

Defines the transfer function as the ratio of output to input in the frequency domain, using impulse response and Fourier transform with inverse transform to analyze system behavior.

Explore how a system's impulse response defines its transfer function and frequency-domain behavior, showing how a delta input yields impulse output and how time-domain convolution corresponds to frequency-domain multiplication.

Explore impulse response, transfer function, and step response, and examine how convolution, commutative and distributive properties govern deconvolution and parallel connections of system impulse responses.

Explore the associative property of LTI systems using impulse responses and block diagrams. Relate input and output via convolution of impulse responses to obtain the overall system impulse response.

Explore memory and memoryless, static and dynamic systems using impulse-response criteria, determine causality from the impulse response, and assess stability through finite impulse response integral for continuous and discrete cases.

Investigate lit systems with properties, focusing on input–output relations, impulse response, inverse systems, and convolution and deconvolution to determine the inevitable system.

Explore the Fourier series introduction, defining a periodic signal as a linear combination of infinite orthogonal functions. Compare trigonometric and complex exponential expansions, and explain orthogonality conditions.

Explore the orthogonality of vectors using dot products and projections. Show that when vectors are perpendicular, the projection is zero and the associated error is minimized.

Learn how signal orthogonality minimizes error energy by projecting signals onto orthogonal components and zeroing the cross-energy, including the complex conjugate extension for complex signals.

Explore how mutually orthogonal signals form a d-dimensional orthogonal signal space and approximate a signal with orthogonal functions, minimizing the error energy.

Explore how the mean square error in signal approximation using a complete set of orthogonal functions drops as more basis functions are included.

Explore how infinite set of mutually orthogonal signals, including sine, cosine, and complex exponentials, can be normalized to unit energy to make mean square error approach zero in signal approximation.

Explore how complex exponential functions form a complete, mutually orthogonal set, with orthogonality shown by zero integrals of their products over [0, T] for distinct indices.

Examine orthogonality in complex functions through inner products, where the integral of a function times the conjugate of another equals zero for orthogonality, and derive coefficients for approximating a function.

The lecture derives the expansion for a full-wave rectified signal, computes its average value, and derives its coefficients via integration of exponential and cosine terms over one period.

Explore Dirichlet's conditions for Fourier series: a periodic, single-valued signal with finite discontinuities and extrema, and a finite integral over one period. These suffice for convergence, but are not necessary.

Explore the complex exponential Fourier series expansion technique for signals, derive coefficients through integration, and verify how even harmonics influence the TFS and EFS representations.

Explore symmetry conditions in periodic signals, including even, odd, and quarter symmetry, to identify when non-zero harmonics occur. Use half-period integration to reduce calculations and save time in exams.

Explore how to check even, odd, and half-period symmetry in signals using time shifts, reflections, and inversion, and determine oximetry and harmonic implications for examples.

Explore exponential signals and their complex representations, combine decaying exponentials, perform integration, and analyze real and imaginary parts using complex conjugates.

Analyze a periodic series of triangular signals and derive its complex Fourier series coefficients, including the dc value, using the complex exponential form.

Analyze how to construct the frequency spectrum for a periodic signal using complex exponentials and magnitude values on the frequency axis, revealing a discrete spectrum.

Explore GATE previous problems with solutions, analyzing periodic signals through Fourier series, dc components, symmetry, and harmonic content to identify the fundamental frequency.

Analyze GATE previous problems on periodic signals, Fourier series, and harmonic content. Explore symmetry conditions such as even and odd, and power ratios of seventh and fifth harmonics.

Explore delta functions and impulse trains, apply shifting properties and one-period integration to analyze impulse expansion and its relation to the signal's fundamental period.

analyze time-shifting effects and exponential decay in signals from the 2015 paper, and explore periodic signals, direct conditions, and Fourier expansion from the 2014 questions.

Analyzing digital conditions for signals, the discussion covers absolute integrability over one period, finite maxima and minima, finite discontinuities, and periodic signals with integration limits.

Examine half-wave symmetry in periodic signals, including time-shift by half the period and inversion, and how these conditions influence signal behavior and Fourier expansion.

Explore symmetry properties of signals, including odd, even, and half-wave symmetry, and determine how these conditions influence periodicity and harmonic content in signals.

Explore in signals and systems how Fourier transforms link time and frequency domains and apply Dirichlet conditions to ensure transformability, including absolute integrability, finite discontinuities, and finite extrema.

Learn how to compute the Fourier transform of the unit impulse using the shifting property, and derive the one-sided exponential spectrum, including magnitude and phase.

Derive the Fourier transform of a two-sided exponential and examine its real, imaginary-free magnitude spectrum, linking time-domain behavior to the frequency-domain profile.

Learn the Fourier transform of the signum function by modifying a signal that is not absolutely integral, splitting the integral, and deriving the final expression in terms of omega.

Learn to express the unit step function using a complex exponential form and derive its Fourier transform, then apply similar steps to sinusoidal functions to understand their frequency-domain representations.

Derive the Fourier transform of a rectangular function, revealing its sinc form and its link to the sampling function via duality and delta properties.

Apply Fourier transform properties to triangular and trapezoidal functions, using ramp, step, and delta functions and differentiation, to derive the transform as (sin(ω/2)/(ω/2))^2 and connect to sampling function.

Analyze the Fourier transform of a trapezoidal signal by decomposing it into ramp and step functions, deriving its second derivative as delta functions, and expressing the transform with exponentials.

Apply the superposition principle and homogeneity to the Fourier transform of a signal A x1(t) + B x2(t), showing F{A x1(t) + B x2(t)} = A X1(ω) + B X2(ω).

Explain the time scaling property of the Fourier transform by showing that scaling a signal in time by a yields (1/|a|) X(ω/a) in frequency, with a not equal to 0.

Learn the time shifting property of the Fourier transform, showing that a time delay corresponds to multiplying the spectrum by e^{-j omega d} using the standard transform form.

Explore the frequency shifting property of the Fourier transform and how time shifts become frequency shifts via multiplication by an exponential in time, as described with omega notation.

Demonstrate and prove the differentiation in time domain property of the Fourier transform, using the standard form and exponential terms.

Explore the integration in time domain property of the Fourier transform, proving how time-domain integration corresponds to frequency-domain expressions and validating the inverse Fourier relationship.

Explore and prove the differentiation in the frequency domain property of the Fourier transform, showing how derivatives with respect to omega relate to the transform and its inverse.

Demonstrates the conjugation property of the Fourier transform by linking the complex conjugate and negative frequency, showing how conjugation affects the transform of a signal.

Explore the duality property of the Fourier transform, showing how a time-domain signal corresponds to a frequency-domain version via omega.

Understand how multiplying a signal by an exponential shifts its Fourier spectrum by plus or minus the modulation frequency, enabling spectral analysis of modulation techniques in analog communications.

Explore how to determine the area under signals in time and frequency domains using Fourier transform properties, including area equivalence across domains and inverse transform insights.

Explore the time-domain convolution and its frequency-domain counterpart, proving that the Fourier transform of a convolution equals the product of spectra, and illustrating time and frequency shifting properties.

Explain the frequency domain convolution property of the Fourier transform, showing that the Fourier transform of a product equals the convolution of the two spectra in frequency.

Parseval's relation proves energy equivalence between time and frequency domains, enabling spectral analysis of signals and random processes in communication systems, via the magnitude squared of the Fourier transform.

Explore how a periodic signal is expressed by the exponential Fourier series and how its Fourier transform becomes a train of delta impulses in the frequency domain.

Explore the Hilbert transform as a linear, time-varying system with impulse response h(t)=1/(π t) that preserves magnitude and adds a ±90° phase shift across frequencies, enabling wideband signal representations.

Explore the Hilbert transform with an example, using impulse sampling and Fourier-domain analysis to derive the transform, transfer function, and frequency-domain relationships.

Explain how to form the frequency spectrum from a signal by separating magnitude and phase via the Fourier transform, and analyze real and imaginary parts and symmetry with examples.

Analyze GATE previous problems on signals and systems, deriving outputs from impulse response via convolution, applying time-shift properties and Fourier transforms, including rectangular frequency responses.

Explore gate-style problems on energy calculation and Fourier transform properties, converting signals between time and frequency domains, applying shifting, scaling, and rectangle function concepts.

Explore fundamentals of digital signal processing, including Laplace transform concepts, rectangular, triangular, and impulse functions, and duality properties for transfer analysis.

Explore the delta function and its Fourier transform, then apply the time scaling property to show how compression in time yields expansion in frequency, and vice versa.

Explore how a real and even time-domain signal yields an odd or imaginary frequency-domain spectrum, and apply frequency shifting and delta function properties to infer inverse transforms and time-domain behavior.

This lecture analyzes the delta function's role in mapping the time-domain impulse to the frequency domain, highlighting symmetry and the frequency shifting property with inverse transforms.

Explore impulse response, input-output relations, and the convolution–multiplication principle between time and frequency domains, and apply time-scaling properties to analyze signals and systems.

This lecture analyzes the relationship between the step function and its Fourier transform, using duality to derive time- and frequency-domain representations and identify the transfer function components.

The lecture explains the frequency shifting property in Fourier transforms, showing how multiplying a time-domain signal by an exponential shifts its spectrum, and derives the transform of a decaying exponential.

Explore the Laplace transform of the impulse (delta) function, applying the shifting property and ROC considerations to determine convergence and transform results.

Examine the Laplace transform of the unit step function and its region of convergence, establishing that the transform exists for Re(s) > 0 and represents a right-sided signal.

Explore the Laplace transform of the left-sided unit step function, identify its region of convergence as sigma less than zero, and compare transforms and plots to right-sided cases.

Compute the Laplace transforms of exponential functions and analyze their region of convergence. Explore right-sided and left-sided signals and their convergence conditions.

Derive the Laplace transforms of complex exponentials and of cos and sin functions, using the region of convergence to relate sigma and omega in the transform.

Explore the Laplace transforms of left- and right-sided exponentials and determine their regions of convergence, showing how poles and sigma conditions define these regions.

Analyze the Laplace transform and region of convergence of a damped sine function, derive its complex exponential form, identify poles, and explain real-part constraints that determine the ROC.

Derive the Laplace transform of a damped cosine, showing L{e^{-a t} cos(ω t)} = (s + a)/[(s + a)^2 + ω^2], with a ROC of Re(s) > -a.

Derive the Laplace transform of hyperbolic sine and cosine functions and establish their region of convergence, using s and omega in the analysis.

Proving the linearity property of the Laplace transform: L{a x1 + b x2} = a X1 + b X2 using the standard integral definition.

Demonstrate the time-shifting property of the Laplace transform, including a proof and derivation showing how time shifts convert to an exponential factor in the transform.

Demonstrate the frequency shifting property of the Laplace transform and how s-domain shifts correspond to multiplying by exponentials in time, relating to the time shifting property.

Explore the time scaling and time reversal properties of the Laplace transform and how time-domain manipulations reflect in LT expressions.

Explore the time differentiation property of the Laplace transform and its proof for the unilateral transform, showing how differentiation in time connects to the Laplace domain and X(0).

Explore the differentiation in s-domain property of the Laplace transform, proving that the right-hand side derivative matches the Laplace transform of the left-hand side expression.

Explore the conjugation property of the Laplace transform, using complex conjugates to show that the transform of a complex-conjugated signal equals the complex conjugate of the original transform.

Explore the Laplace transform's initial value data and final value data, and the initial value theorem and final value theorem, proven via the differentiation property, to link time-domain signals.

Demonstrate the convolution property of the Laplace transform by converting a time-domain convolution into a product of Laplace transforms, X1(s) and X2(s).

Explore gate-style problems on Laplace transforms and transfer functions, deriving impulse responses and system outputs from inputs using time-domain shifting and inverse Laplace transforms.

Explore solving GATE previous problems using Laplace transforms, convolution properties, and time-domain shifting with unit step and rectangular functions to obtain explicit time-domain results.

Apply Laplace transforms to a differential equation with zero initial conditions, use partial fractions to simplify the transfer function, and derive the time-domain response e^{-t} - e^{-3t}.

Analyze a GATE problem on a causal continuous-time system: determine the transfer function, derive impulse response via partial fractions, and verify steady-state with a unit step input.

Explore signals and systems concepts through a GATE problem set, deriving transfer functions from impulse and step responses using convolution and Laplace transforms.

Analyze a second-order system's impulse response via its differential equation and transfer function, identify poles at -2 and +3, and determine stability and causality using left-sided and right-sided impulse responses.

Explore solving gate previous problems with solutions set 7 using Laplace transforms, verify expressions, determine alpha and beta, and compute the area under a given function via Laplace-domain techniques.

Master GATE previous problems on linear time-invariant systems by analyzing input-output relations, impulse responses, deconvolution, and Laplace-transform techniques with RAM functions and time-domain shifts.

This lecture covers GATE previous problems set 9 on signals and systems, teaching how to obtain input from output via impulse response and deconvolution, and applying the initial value theorem.

Explore how to compute Laplace transforms of signals using time shifting, exponential modulation, and the differentiation property, and apply convolution and standard transform pairs.

Explore solving a circuit using Laplace transforms, applying initial conditions, constructing a transfer function via partial fractions, and performing inverse transforms with s-domain shifting and exponential terms.

Derive the system transfer function from input-output data using impulse response and the Laplace transform, and apply shifting and convolution in the Laplace domain.

Examine how signals respond to time shifts and exponential modulation, plot rectangular functions, and apply the initial value and final value theorem to determine a system's response.

Convert signals into sums of known functions using time-domain shifts and unit step components, then compute their Laplace transforms with exponential factors.

The lecture analyzes three time functions, including a cost function and a sinusoid with phase alpha, and derives their Laplace transforms using complex exponentials, yielding 1/(s^2+omega^2) type forms.

Analyze the z-transform of the unit impulse and unit step, derive their region of convergence, and express the geometric series as 1/(1-x) under x<1.

Analyze z-transform and region of convergence for left-sided unit-step variants u(-n) and u(-n-1), including shifting and reflection, and show that the ROC corresponds to the interior of the unit circle.

Explore the z-transform of exponentials a^n u(n) and -a^n u(-n-1), derive their regions of convergence, and visualize ROC in the z-plane for stable analysis.

Compute the z-transform of complex exponentials and cos ωn u(n) by expressing cos ωn as a pair of complex exponentials, and analyze magnitude and ROC outside the unit circle.

Explore the z-transform and region of convergence for sin(ω n) u(n) by expressing the sinusoid with complex exponentials and deriving the transform properties.

Explore the linearity property of the z-transform, showing that the transform of a weighted sum equals the weighted sum of individual transforms, with X1 and X2 representing signal transforms.

Explore the time shifting property of the z-transform, deriving left and right side expressions and showing how time-domain bounds from minus infinity to plus infinity relate to shifted signals.

Examine the third property: multiplication by an exponential sequence in the time domain and its corresponding effect in the transform domain, with a proof of the equality.

Explore the time reversal property of the z-transform, showing how reversing a signal affects its transform, using the standard transform formula and limits from minus infinity to plus infinity.

Explain the time expansion property of the z-transform, derive its time-domain implications for a given signal, and relate left and right side expressions in the p-domain.

Explain and prove the z-transform differentiation property by differentiating X(z) with respect to z and evaluating limits to connect to the time-domain operation.

Explain the conjugation property of the jet transform, showing that the complex conjugate of the transform equals the transform of the complex conjugate and how powers interact.

Master the convolution property of the z-transform, showing how time-domain convolution of two signals corresponds to the product of their transforms.

Explain the z-transform initial value and final value theorems for causal signals, using limits to infinity and zero to validate them.

Explore z-transform properties and the final value theorem for causal signals, emphasizing time-shifting and non-zero initial conditions. Derive how the final value links to the transformed sequence.

Solve gate-style problems in signals and systems, covering causal and non-causal signals, unit circle stability, and initial- and final-value theorems.

Analyze discrete-time system stability using transfer functions and pole locations inside the unit circle. The lecture works through GATE-style problems, deriving transfer functions and stability conditions for causal systems.

Explore GATE previous problems with solutions by applying time-domain and transform techniques to signals, and analyze transfer functions with poles, zeros, and imaginary-axis behavior.

Analyze causal transfer functions and impulse responses, determine stability criteria, and derive A and B values for stable discrete-time systems; discuss two-sided signals and boundary reasoning in the IES discussion.

Analyze finite duration sequences in the time domain and their transforms, focusing on convergence as values approach infinity, and apply the approach to computing x(1/2) in a 2015 IES discussion.

Examine stability criteria using margin conditions, analyze pole relations, and apply the final value theorem to a rational function to determine X infinity.

Derive step responses from transfer functions and analyze their time-domain and z-domain representations, while solving for system constants via differential and difference equations.

Analyze the system's transfer function as the output-input ratio, derive it from the differential equation using the shifting property of transforms, and connect continuous and discrete time signals.

Explore the discrete time Fourier transform (DTFT) as the discrete counterpart to the continuous Fourier transform, using complex exponentials and summation to analyze discrete-time signals.

Explore the dtft of impulse and unit step signals, showing the delta function has a dtft of one and the unit step transforms to 1/(1−e^{−jω}).

Explore the discrete-time Fourier transform of a discrete-time exponential sequence by expressing it as a geometric series and applying the transform formula, confirming equivalent representations.

Explains the discrete Fourier transform, linking a discrete time-domain signal to its continuous frequency-domain representation, and presents the DFT formula and its inverse on a computational grid.

The lecture walks through a four-point dft example, deriving x[0] to x[3] and revealing that x[1] and x[3] form complex conjugates, with verification via the symmetry relation.

Explore gate level signals and systems by solving discrete-time sequence problems using complex exponentials and transform techniques through set 1 with detailed solutions.

Explore GATE previous problems with solutions set 2 on discrete Fourier transform properties, including complex conjugates, observation points, and nonzero DFT coefficients for multi-point sequences.

Define the sampling theorem and Nyquist condition, explain time-domain and frequency-domain analyses of sampling, and illustrate reconstruction from uniformly sampled data at twice the highest frequency.

Analyze the Nyquist condition and NR calculations by identifying the highest frequency components, convolving in the frequency domain, and using Fourier transform of rectangular functions to relate time and frequency.

Explore time domain and frequency domain analysis of sampling, Nyquist condition, and reconstruction of band-limited signals using delta train sampling and low-pass filtering.

Explore how sampling rate and reconstruction filters shape output spectra, identify frequency components, and apply band-pass and low-pass filters using convolution and Fourier transforms.

explain distortionless transmission systems by deriving the transfer function, showing a constant magnitude and a linear phase corresponding to delay, and obtaining the impulse response from the transfer function.

Derive the impulse response from a distortionless transmission system's transfer function using Fourier transforms, and show that the ideal low-pass filter yields a non-causal impulse response, making it physically unrealizable.

Explore how linear systems act as frequency selectors, shaping input power spectra through the transfer function, with low pass, high pass, band pass, and band rejection types.

Explore signal bandwidth and system bandwidth, defining signal bandwidth as the range of positive frequency components, and show system bandwidth should exceed signal bandwidth to prevent distortion.

Explore convolution with unit step and impulse response signals through concrete examples, showing shifting, reflecting, and multiplying signals to yield outputs like parabolic function, and review convolution properties.

Learn to perform graphical convolution of an exponentially decaying signal with a shifted unit step, using time shifting, time reversal, and multiplication to obtain the convolution result.

Explore convolution of two rectangular signals using a graphical approach, shifting and reflecting one signal, yielding a trapezoidal to triangular result when the weights are equal.

Convolve a triangular function with a rectangular function to derive the resulting piecewise signal, using graphical methods and time limits analysis.

Explore correlation of signals, defining cross-correlation and auto-correlation, using shifting and orthogonality to measure similarity, and relate auto-correlation to the power spectral density.

Clarify the relation between convolution and cross-correlation by showing that cross-correlation equals convolution with the second signal reversed, and that if one function is even, they are equivalent.

Explore how a linear system's transfer function, via the Fourier transform, controls input and output spectral densities, with output PSD equal to input PSD times the squared magnitude of H(jw).

Explore the graphical evolution of cross-correlation between triangular and rectangular signals, highlighting shifting, area-based multiplication, and common areas to derive the correlation function.

Examine how the power spectral density relates to a signal's average power by integrating PSD across frequency in piecewise ranges -4 to -2, -2 to 2, and 2 to 4.

Explore the relation between the auto correlation function and the power spectral density using Fourier transforms, and examine their symmetry and power properties.