Undergraduate course on signals and systems(Course-II)

Explore how the continuous time Fourier transform (fft) extends analysis beyond periodic signals, converting time-domain signals into a frequency-domain spectrum and revealing time-frequency duality.

Explore how to derive the continuous-time Fourier transform from the continuous-time Fourier series by converting a finite-duration signal into a periodic one and applying synthesis and analysis equations.

Derive the continuous-time Fourier transform by letting the frequency step go to zero, turning a discrete sum into an integral and establishing x(t) and X(ω) as a Fourier transform pair.

Analyze how x[n] maps to spectral components at multiples of ω0 as T0 changes ω0. Doubling T0 increases spectral density and shapes the envelope X(jω) for continuous-time Fourier series.

Examine the Fourier transform pair x(t) and X(ω), and distinguish analysis and synthesis equations that map between time and frequency domains using the continuous-time Fourier transform and its inverse.

Explore area calculations for time-domain signals and their spectra using the continuous-time Fourier transform. Relate zero-frequency areas: time-area equals X(0); spectrum-area equals 2π x(0).

Analyze how real or imaginary time signals with even or odd symmetry determine the CTFT; real and even yield real, even spectra, others yield imaginary or odd spectra.

Understand symmetrical conditions that link time-domain parity to spectrum. Real and even time-domain signals yield real or imaginary spectra, while odd components swap real and imaginary in the frequency domain.

Understand the weak Dirichlet condition and absolute integrability as criteria for Fourier transform existence. Note that energy signals have a Fourier transform, while the unit step remains an exception.

Explore time and frequency shifting properties in the continuous-time Fourier transform, showing time shifts introduce phase factors without magnitude change, and frequency shifts correspond to time-domain modulation by exponentials.

Explore time scaling: compression in time corresponds to expansion in the frequency domain, while a>1 expands time and compresses the frequency domain; inversion in time yields inversion in frequency.

Highlight time differentiation and time integration properties in Fourier analysis, showing how differentiating x(t) yields jω x(ω) and integrating yields x(ω)/(jω) with a delta term in ω and in f-domain.

Explore the linearity and convolution properties of signals, showing how time-domain operations map to frequency-domain effects and how convolution equals multiplication in the Fourier domain.

Discover duality in the Fourier transform, interchanging time and frequency to show that if x(t) ↔ X(ω), then X(t) ↔ 2π x(-ω); a universal, pair-based property for all signals.

Explore Parseval's energy theorem, showing that the energy of a time-domain signal x(t) equals one over two pi times the energy of its spectrum, via the Fourier transform.

Derive Fourier transform of unit impulse delta(t) as a constant in frequency, then apply duality to show a constant in time corresponds to 2 pi delta(omega), forming a transform pair.

Analyze the Fourier transform of real exponential signals with a>0. The FT of e^{-a t} u(t) equals 1/(a + j omega), with magnitude 1/sqrt(a^2 + omega^2) and phase -tan inverse(omega / a).

Derive the Fourier transform of the real exponential x(t)=e^{-a t} u(-t), yielding X(ω)=1/(a−jω) with magnitude 1/√(a^2+ω^2) and phase tan inverse(ω/a); show negative-time support and its spectra.

Explore the Fourier transform of the real exponential signal e^{-a|t|}, derived by splitting into e^{-a t}u(t) and e^{a t}u(-t); the transform is 2a/(a^2+omega^2), with a real, even spectrum.

Derive Fourier transform of signum function by expressing it as a difference of Heaviside steps, yielding a spectrum 2/(jω) that is purely imaginary and odd, with 1/(jπ f) in f-domain.

Explore the Fourier transform of the unit step, its relation to signum, and how a partial term plus delta impulse capture the jump and constant behaviors, despite Dirichlet conditions.

Derive the time integration property from the Fourier transform of the unit step signal, showing x(t) convolved with u(t) equals X(ω)/(jω) + π X(0) δ(ω).

Explore the Fourier transform of a rectangular pulse rect(t/tau), a finite energy, even signal with width tau from -tau/2 to tau/2, yielding sin(omega tau/2)/(omega/2) form.

Examine the frequency-domain sampling function for real, even signals, noting a real spectrum with zeros at omega = ±k pi and its relation to rect and sampling via Fourier transform.

Explain how the Fourier transform of a rectangular function yields a sinc function, illustrating the rect-sinc pair in time and frequency domains with related sampling relationships.

Explore how the Fourier transform of a triangular pulse arises from convolving two equal-width rectangles, yielding a squared rectangular spectrum in omega.

Apply the duality property to derive the Fourier transform of sampling square t as (pi/2) triangle(omega/2). The lecture links the time-domain sampling square t to its even triangular spectrum.

this lecture shows that the Gaussian pulse e^{-π t^2} is its own Fourier transform, becoming e^{-π f^2} in the frequency domain, derived using time and frequency differentiation properties.

Showcases Fourier transform of complex exponentials and sinusoidal signals via frequency shifting, mapping e^{-j ω0 t} and e^{j ω0 t} to 2 pi delta(ω ± ω0) impulses.

Express cos ωt via Euler's formula and derive its Fourier transform with delta impulses at ±ω0 and magnitude π, then outline sine's transform and phase.

Learn how the Fourier transform applies to periodic signals, yielding a spectrum of impulses at multiples of the fundamental frequency with weights equal to Fourier series coefficients.

Compute the Fourier transform of a periodic impulse train with period four, using x_n = 3/4, yielding impulses of strength 3π/2 at multiples of ω0 where ω0 = π/2.

Learn how to analyze LTI systems with the impulse response h(t) using the Fourier transform. Convolution in time becomes multiplication in frequency, giving y(jω)=x(jω)h(jω) and revealing magnitude and phase responses.

Distortionless transmission yields an exact replica of the input, aside from fixed delay and gain, with constant magnitude and linear phase, H(jw)=k e^{-j w tau}.

Explore how a frequency selective LTI system defines filters through spectrum representations. Learn about low pass, high pass, bandpass, band reject, and allpass filters with cutoff frequencies omega_l and omega_h.

Explore ideal sampling bridging continuous-time and discrete signals. Ensure recovery by using a sampling frequency at least twice the maximum frequency; examine baseband, bandpass, impulse trains, and the frequency-domain view.

Analyze ideal sampling of baseband signals by impulse train modulation, revealing spectral replicas spaced by omega_s and three cases: omega_s > 2 omega_m, = 2 omega_m, and < 2 omega_m.

demonstrates how sampling at twice the maximum frequency creates replicas and possible aliasing; explains cases with omega_s equal to two omega_m and omega_s less than two omega_m, highlighting spectral overlap.

The lecture compares oversampling, Nyquist sampling, and undersampling, showing how X(Ω) can be recovered from oversampling and Nyquist cases using practical or ideal low-pass filters, while undersampling causes aliasing.

Natural sampling replaces the impractical ideal impulse train with a small rectangular pulse train from a stable multivibrator, gated by a MOSFET switch to sample x(t).

Learn flat top sampling as a natural sampling extension with a sample-and-hold capacitor that captures a maximum value per delta duration and retains it, yielding flat tops.

Explore zero-order hold interpolation for reconstructing continuous-time signals from samples, using an LTI system with a rectangular impulse response of width equal to the sampling period.

First order hold interpolation smooths discrete samples by cascading two zero order holds, yielding a smoother recovery of x(t) from x(nTs) via convolution with h(t), though perfect recovery remains impossible.

Learn how the Laplace transform overcomes Fourier limits by converting a broader class of exponential signals, including positive exponentials, into the frequency domain via eigenfunctions.

Learn to derive the Laplace transform from the Fourier transform using an exponential, define s = sigma + j omega, and relate x(t) to x(s) via analysis and synthesis equations.

Explore Laplace transform pair, linking the time-domain signal x(t) to its Laplace transform x(s), and compare with Fourier series and Fourier transform, using the partial fraction method for inverse transform.

Explore unilateral and bilateral Laplace transforms: unilateral for causal signals in solving differential equations with initial conditions; bilateral handles all signal types with region of convergence and stability criteria.

explains the pole-zero representation of Laplace transforms by defining zeros as numerator roots and poles as denominator roots on the s-plane, and shows how these determine ROC and stability criteria.

We explain the existence condition for the Laplace transform, showing that the integral of |x(t)| e^{-sigma t} dt must be finite for bilateral or unilateral transforms.

The region of convergence governs bilateral Laplace transforms; signals e^{a t} u(t) and -e^{a t} u(-t) share 1/(s-a) but have ROC sigma > a and sigma < a, respectively.

Explore the progression from continuous-time transforms to discrete-time transforms, connecting Fourier, Laplace, and the jet transform, and discuss convergence in solving LTI systems and difference equations.

Relate the z-transform to the Laplace transform by comparing discrete x[n] with the complex variable z, expressing X(z) as a summation of x[n] z^{-n} and its inverse via contour integral.

Map the s-plane to the z-plane via z = e^s. The j omega axis maps to the unit circle; left half (sigma<0) maps inside, right half (sigma>0) maps outside.

Analyze the bilateral z-transform by expressing X(z) as N(z)/D(z) and determine the region of convergence on the z-plane using poles and zeros, then apply stability criteria to LTI systems.

The unilateral z-transform uses a lower limit of zero and applies only to causal signals. It yields a unique X(z) and solves constant coefficient difference equations without requiring ROC.

Determine the condition for existence of the jet transform by analyzing X(z) and its convergence. Show that |x[n]| r^{-n} sums must converge, giving bilateral and unilateral cases.

Explore the region of convergence in the z-transform by comparing two signals with the same transform, one causal and one anti-causal, yielding ROC |z|>|a| and |z|<|a|.

Identify the z-transform region of convergence for causal and anti-causal signals using a unit circle reference, and assess dtft existence and stability by whether the unit circle lies in roc.

Explore the region of convergence properties for the jet transform, noting poles lie outside rings around poles; classify signals as infinite or finite duration with right-, left-, and two-sided types.

An infinite duration left sided signal comprises three parts multiplied by shifted unit steps and extends toward minus infinity; the roc is the intersection of the rocs inside innermost pole.

Examine infinite duration two-sided signals by intersecting left- and right-sided rocs in the z-transform, revealing a ring-shaped convergence region or, when empty, no roc.

We examine a finite duration right-sided signal with samples from n=0 to 4 and show its z-transform has four poles at z=0, so ROC is the entire z-plane except z=0.