Undergraduate course on Signals and Systems(course-I)

Explore how a finite-duration signal creates a periodic signal by shifting it into infinite replicas and summing them, revealing the fundamental period and the repeating pattern.

A periodic signal repeats a pattern with a fundamental period T0, so shifting by T0 yields no change. The fundamental frequency equals 1/T0, f0, and ω0 equals 2π/T0.

Describe discrete-time periodic signals where X(n) repeats every N0 samples and satisfies X(n)=X(n±kN0). Define the fundamental frequency ω0=2π m/N0, a rational multiple of 2π.

Determine the periodicity of a sum of periodic signals by ratio analysis, distinguishing rational and irrational cases, then compute the fundamental period using LCM and common-factor elimination.

Determine the periodicity of the sum of three periodic signals by analyzing fundamental period ratios, then compute the fundamental period for each set using LCM and ratio simplification.

Study sinusoidal signals as the basis of periodic signals, covering continuous-time and discrete-time domains, the 90-degree phase difference between sine and cosine, and generation with an oscillator.

Derive a discrete-time sinusoid from a continuous sine by sampling eight samples per cycle, producing a repeating sequence with omega = pi/4.

Explore complex exponential signals in continuous and discrete time, using e^{j ω t} and its conjugate e^{−j ω t} to connect to cos(ω t) and sin(ω t), noting periodicity.

Compare continuous-time and discrete-time sinusoidal signals by periodicity: continuous-time sinusoids are periodic for nonzero omega, while discrete-time sinusoids are periodic only if fundamental frequency is a rational multiple of 2π.

In continuous time, a sinusoid has a unique fundamental frequency. In discrete time, signals separated by multiples of 2π are identical, giving many frequencies the same sequence.

Compute the fundamental periods of three sinusoids, check their ratios for periodicity, and use the lcm to obtain the overall fundamental period; the example yields 140.

Analyze the periodicity of the sum of three sinusoidal signals by computing fundamental periods pi/3, pi/4, and pi/12, and verify periodicity via rational ratios to obtain the fundamental period.

Analyze a mixed example of three sinusoidal signals to determine their fundamental frequencies and periods, and explain how pi-related ratios influence the periodicity of their sum.

Identify fundamental frequencies pi/5, 2pi/7, and 3pi/70 in a discrete-time signal, compute their fundamental periods, and obtain the overall fundamental period via the LCM; compare with a direct method.

Explore the periodicity of discrete signals x2[n] formed by cos(3n/5) and sin(5n/3). Show both components are periodic, their sum is periodic, and note cases where the fundamental period is infinity.

Learn to determine fundamental frequencies and periods of discrete-time signals using product-to-sum identities and complex exponentials, yielding periods 24 and 72 in the examples.

Explore real exponential signals in both continuous-time and discrete-time domains, using x(t)=K e^{a t}. Cover the cases a=0, a>0, and a<0, showing constant, positive exponential, and negative going exponential behaviors.

Explore how real exponential signals arise in the discrete-time domain by sampling a continuous signal, with beta times gamma^n, covering positive, negative, and constant cases and convergence behavior.

Identify energy signals and power signals with finite energy or finite power, and classify signals as neither, converging or diverging, with examples like sinusoidal and unit step.

Compute instantaneous power as v(t) i(t) for a one-ohm resistor, giving p(t) = x^2(t). Energy is the integral of p(t); power is its time average.

Explore energy and power expressions for real and complex signals across continuous time, continuous frequency, and discrete time domains, using modulus in the energy and power formulas.

Master the root mean square value for periodic signals by understanding its definition as the square root of average power, with the formal expression using time average of x(t)^2.

Analyze how shifting, inversion, and scaling affect an energy signal and its power signal; shifting and inversion leave energy and area unchanged, while scaling changes energy to E{x(at)} = E{x(t)}/|a|.

Study how time shifting, scaling, and time inversion do not change the average power of a power signal. The lecture shows the average power remains constant under these operations.

Compute the average power of sinusoidal signals and show it equals the square of the peak value divided by two for cosine and sine, independent of frequency and phase. When combining multiple sinusoids, the average power is the sum of the individual powers.

Apply the superposition principle to compute the average power of a dc term plus sinusoidal and complex exponential components, using modulus-squared amplitudes for the exponentials and half-squared for the sinusoids.

Compute the energy delivered by a 12 V battery at 2 A as voltage drops to 10 V over 10 minutes, by integrating power to obtain 13.2 kilojoules.

Compute the rms value of a periodic signal by first calculating its average power over one fundamental period, then take the square root.

Compute the amplitude of the signal 3+4cos(3t) by separating its dc and sinusoidal parts. Average power is 9 for constant and 8 for cosine, totaling 17, giving amplitude sqrt(17).

This lecture shows that shifting and scaling a signal's time axis do not affect its average power, via the relation between x(t) and gft and the fundamental period.

Define and compare even and odd signals in real and complex domains using time-inversion and amplitude-inversion criteria. Demonstrate continuous-time and discrete-time examples of conjugate symmetrical and anti-symmetrical behavior.

Demonstrates that a real signal not purely even or odd can be expressed as a sum of its even and odd parts, using x_e(t)=(x(t)+x(-t))/2 and x_o(t)=(x(t)-x(-t))/2.

Explore the area of even signals and zero area of odd signals in both continuous and discrete time. In discrete time, even signals double the sum and odd signals cancel.

Explore conjugate symmetric and conjugate antisymmetric signals, noting that the real part is even and the imaginary part is odd for symmetric, with the reverse for antisymmetric.

Explains conjugate symmetric and conjugate anti symmetric properties for complex signals and shows their decomposition into x_c[n] = (x[n] + x*[−n])/2 and x_ca[n] = (x[n] − x*[−n])/2.

Solve for the even and odd components of a conjugate symmetric or antisymmetric signal x(t)=e^{-2t}cos t, using x_e(t)=(x(t)+x(-t))/2 and x_o(t)=(x(t)-x(-t))/2, yielding x_e(t)=cos t cosh(2t) and x_o(t)=-cos t sinh(2t).

Compute the even and odd parts of x1(t)=cos t+sin t+sin t cos t using x_e=(x1+x1(-t))/2 and x_o=(x1-x1(-t))/2; obtain x_e(t)=cos t and x_o(t)=sin t+sin t cos t.

Learn to decompose a multi-component signal into even and odd parts using the shortcut (x(t)+x(-t))/2 and (x(t)-x(-t))/2, applied component-wise to constant, cos, and sine-cosine terms.

Demonstrates obtaining the conjugate antisymmetric part of a finite sequence by time reversing x(n), conjugating, and computing a(n) = [x(n) − x(−n)*]/2, with support from −1 to 1.

Explore singularity functions, including the unit impulse, unit step, and unit ramp signals in continuous and discrete time, with their unit-area Dirac delta representations and definitions.

Explore how the unit impulse in the continuous time domain is defined by its area, with rectangular pulses illustrating unity area despite changing width and height.

Study the sampling property of unit impulses in continuous-time and discrete-time signals: multiplying by a delta yields an impulse whose area equals the signal value at the impulse location.

Explore the sampling property in the discrete-time domain by multiplying a sequence x[n] with unit impulses delta[n−k]. Each result is x[k] delta[n−k], showing left and right shifts via delta[n+K].

Apply the sifting property of the unit impulse in continuous and discrete time to extract X at the impulse location, e.g., X(0) or X(-K).

The lecture covers the scaling property: continuous-time compression by a reduces a rectangular window width to 2/a and area to 1/a, approaching delta; discrete-time compression preserves delta[n].

Explore the differentiation property of the continuous-time unit impulse, introducing the doublet (delta dash) function and showing that the integral of x(t) delta dash(t) equals minus x'(0).

Explore impulse trains in continuous and discrete time, showing unit impulses at regular intervals form a periodic signal and presenting their representations as sums of shifted delta functions.

learn that any discrete-time signal can be expressed as a weighted summation of shifted impulses, with x[n] = sum_{k=-infinity}^{infinity} x[k] delta[n-k], where the amplitudes are the weights.

Learn how any continuous time domain signal can be represented as a weighted integration of shifted impulses using delta functions, blending analysis and synthesis.

Apply the sifting property of the unit impulse to evaluate continuous-time integrals with delta functions. Compute the integral using x(t)=e^{-2t} and obtain e^{-2}.

Learn to evaluate integrals with delta impulses using the sifting and shifting properties. Distinguish finite from infinite limits to decide if the impulse yields zero or x(3)=12.

Solve integrals using shifting and scaling of delta impulses with the sifting property, then apply the differentiation property to delta dash in a signals and systems context.

apply the sifting property to a shifted impulse at t=4 with a rectangular signal x(t) that equals 1 for |t|≤2, yielding the integral value 2.

Define the unit step signal in discrete and continuous time, equal to one for n ≥ 0 and zero for negative values, with a jump discontinuity at zero, ideal signal.

Explore the practical continuous-time unit step signal with finite transition times and linear ramps, contrast the ideal discontinuity, 3 u(t) amplitudes, and impulse signal area.

Examine how scaling a unit step preserves behavior in continuous and discrete time, including compression, expansion, and the integer condition for right-shifting with B and A.

The unit step signal has infinite energy and a finite average power of one by two, revealing it as a power signal.

Explore how a unit step signal can be decomposed into its even and odd components, with mathematical and graphical representations of ue(t) = (u(t)+u(-t))/2 and uo(t) = (u(t)-u(-t))/2.

Construct a rectangular pulse by subtracting a right-shifted from a left-shifted unit step signal. The width equals the distance between shifts and jump discontinuities reflect the number of steps.

Examine causal, non causal, and anti causal signals and how unit step multiplication converts a non causal signal to causal or anti causal forms, with a jump at t=0.

Explore causal, non causal, and anti causal signals in the discrete time domain by defining amplitudes for n≥0, n<0, and using unit step to obtain causal or anti causal signals.

Analyze causality conditions across continuous and discrete signals, classifying signals as causal, non-causal, or anti-causal, with finite and infinite duration examples.

Explore how multiplying noncausal signals with unit step signals and their shifts yields causal or anti-causal responses in continuous and discrete time, and how rectangular pulses extract signal portions.

Compute the energy of real exponential signals by using a unit step to extract the converging portion; both e^{-a t} u(t) and e^{a t} u(-t) have finite energy equal to 1/(2a).

Explore the energy of real exponential signals by decomposing with U(T) and U(-T), identifying converging and diverging parts, and showing e^{-|a| t} has energy 1/a.

Analyze the energy of real exponential signals in the discrete-time domain by splitting into causal and anti-causal portions; for 0<alpha<1 the energy is 1/(1-alpha^2), and for alpha>1 it is 1/(alpha^2-1).

Explore how the unit step relates to unit impulse in discrete time using first difference (u(n) minus u(n-1)) and in continuous time via the first order derivative (du/dt = delta(t)).

Derive the unit step from the unit impulse in both discrete and continuous time by expressing signals as sums or integrals of shifted impulses, showing accumulation and integration relationships.

Examine how the unit step and unit impulse relate via first difference and differentiation, and how accumulation and integration produce step signals in discrete time.

Define the signum function from the unit step, as u(t)−u(−t) or 2u(t)−1; reveal its odd, periodic power nature and its role in the continuous-time Fourier transform of the unit step.

Show how the even parts of cos(π t) u(t) and sin(π t) u(t) simplify to periodic signals, concluding that both x1(t) and x2(t) are periodic.

derive y(t) as a unit-amplitude rectangular pulse from minus three to plus three by analyzing the shifted impulse structure. compute its energy by integrating y^2 over that interval, yielding six.

Demonstrate expressing a discrete-time rectangular window using step signals, showing X(n) = u(n) − u(n−5) that equals 1 for n = 0 to 4 and 0 elsewhere.

Calculate energies for continuous-time and discrete-time signals. Compute 1/6 for x(t)=e^{-3t}u(t), 1/3 for x(t)=e^{-3t}u(t)+e^{3t}u(-t), and 4/3 for x[n]=(1/2)^n u[n].

Identify four jump discontinuities of x(t) at -2.5, -1, 3.5, and 5; express x(t) as a linear combination of four step signals: 0.75 U(t+2.5) - 3 U(t+1) + 4 U(t-3.5) - 2.5 U(t-5).

The lecture analyzes x(t) as a linear combination of step functions with jumps at -2, -1, 2, and 3, and derives its first derivative as impulses whose areas equal jumps.

Define the unit ramp signal in discrete and continuous time: zero for negatives, amplitude equals time, slope reflects ramp, and causal behavior linked to unit impulse and unit step.

Analyze ramp signals by their slope and show that time scaling and amplitude scaling yield the same result only for ramps, including the equivalence of time inversion and amplitude inversion.

Combine two ramp signals with different slopes and shifts to form complex waveforms, including generation of triangular pulses from ramp combinations.

Learn to build a triangular pulse from ramp signals, using three ramps with slopes 1, -1, and 1. Slope changes occur at -2, 0, and 2, equal to ramp count.

Explore relation among ramp, step and impulse signals via a triangular pulse built from three ramps, showing that the first derivative is a step and the second is an impulse.

Express x(t) as a linear combination of ramp signals and solve for its energy. The calculation shows x(t)=r(t-1)-r(t-2)-r(t-3)+r(t-4) with energy 5/3.

Express the given signal as a combination of unit step and ramp signals, identify jumps at zero and at capital T, and determine ramp slopes and slope changes.

Explore the time-domain relationship between sampling and sinc functions, showing sampling of pi t equals the sinc form and zeros at every integer multiple of pi.

Explore the relation between the discrete-time sinc function and its sampling in both f-domain and omega-domain representations. Derive the standard discrete-time sinc forms and understand their zeros at integer multiples.

Explore the six fundamental system properties, including linearity, time invariance, stability, memory, causality, and invertibility, and how they classify and analyze linear time-invariant systems.

Explore linearity in systems by mastering homogeneity and the additive (superposition) condition; a system is linear when scaled inputs yield scaled outputs and the sum of responses matches.

Visualize linearity in signals and systems through graphical analysis of multipliers and a summing element, showing how input components combine to yield the same output across configurations.

Apply a step-by-step procedure to test linearity by assuming homogeneity, verifying the additive condition via two input components and their outputs, and comparing with the output for the combined input.

Split the input into two parts and compare outputs to test linearity. The first example shows y = x^2 is nonlinear; the second example with an exponential relation demonstrates linearity.

Analyze linearity through input-output relations and zero input and zero state responses in RLC networks. Compare resistor, capacitor, and inductor behavior, and apply transformation analysis with constant-coefficient differential equations.

The lecture analyzes linearity and nonlinearity in signals using periodic signals, sine and cosine components, and complex exponentials, and shows how superposition can fail for certain input combinations.

Demonstrates that differentiator and integrator operations are linear systems under zero initial conditions, and explains capacitor and inductor voltage-current relations as linear elements.

Examine a special case where the homogeneity condition fails, making the system nonlinear. The system extracts the real part of a complex signal, and complex scaling breaks expected output scaling.

Demonstrates time invariance: if a shifted input yields an identical shifted output, the system is time invariant; illustrated with a unit step input producing a ramp output and two configurations.

Explore a simple procedure to test time invariance via the input-output relationship, shifting x(t) and verifying the corresponding output matches the shifted y(t).

Analyze whether systems described by input-output relations are time invariant by comparing shifted inputs and responses, with examples using x squared, x cubed, and shifted signals.

Assess time invariance: y = x^2 is time-variant since y(t−τ) ≠ y shifted. In contrast, y = K x(t) is time-invariant.

Explore time invariance through standard examples with sinusoidal inputs, validating cos and sin based responses under time shifts, and confirm differentiator as a time invariant system.

Examine the integrator's input-output relationship and prove time invariance by constructing a right-shifted input x1(t) = x(t - T) and a corresponding y1(t), showing y(t - T) = y1(t).

Analyzes an integral system with a minus infinity lower limit and a finite upper limit to test time invariance, showing that scaling a limit can make the system time-variant.

Examine linearity and time invariance through system classifications as linear time invariant or not, with examples and notes that differentiator and integrator are LTI.

Characterize stability by ensuring every bounded input yields a bounded output. Distinguish boundedness as finite signal values at all times, else the system is unstable.

Assess stability using bounded inputs, noting finite output for squared systems and unbounded output for exponential growth. Conclude differentiator and integrator are unstable.

Examine how signals and systems input-output relationships define causal, non-causal, and anti-causal systems by showing dependence on present, past, or future inputs.

Differentiate memory-based dynamic systems from memoryless static ones by how outputs depend on present and past inputs, with analog capacitor or inductor and RLC networks, and digital sequential circuits.

Identify invertible systems where the input can be recovered from the output. A 1-to-1 mapping ensures invertibility; one-to-many and many-to-one mappings cause non-invertible behavior, as shown by a square-root example.

Examine right shift by four and left shift by four, and determine invertibility by recovering x(t) from y(t). Show many-to-one mappings in discrete time with y[n]=x[n]x[n-1], indicating non-invertibility.

Explain that the differentiator is non-invertible, unlike the integrator in continuous time. Show that discrete-time compression and expansion eliminate samples, making scaling non-invertible, while continuous-time scaling remains invertible.