Signals and Systems Tutoring

Signals and systems is where electrical engineering turns into mathematics. A circuit, a filter, a sampled sensor, and a radio channel all become the same object: a system that takes an input signal and returns an output. Students often follow every lecture and still stall on exams, because the course has several transforms that look alike. STEM Tutor 360 organizes them around one idea. A complex exponential passes through an LTI system changed only by a scale factor, and each transform is a different way of using that fact.

Sessions cover continuous-time and discrete-time signals, system properties (linearity, time invariance, causality, stability), convolution, Fourier series, the Fourier transform, Laplace and z-transforms, frequency response, filtering, and the sampling theorem. We follow Oppenheim and Willsky, Lathi, or your professor's notes, and use the notation your exams use. MATLAB or Python plots come in whenever they help you see what a transform is actually doing to a signal.

A typical convolution problem, worked

Find $y(t) = x(t) * h(t)$ when $x(t) = e^{-2t} u(t)$ and $h(t) = u(t) - u(t - 1)$, a unit-height pulse from 0 to 1.

Write $y(t)$ as the integral of $x(\tau) h(t - \tau)$ over $\tau$. The flipped pulse $h(t - \tau)$ equals 1 for $t - 1 < \tau < t$, and $x(\tau)$ is nonzero only for $\tau > 0$. Slide $t$ from left to right and watch how the overlap changes. That gives three cases.

For $t < 0$ there is no overlap, so $y(t) = 0$. For $0 < t < 1$ the overlap runs from 0 to $t$, so $y(t)$ is the integral of $e^{-2\tau}$ from 0 to $t$, which is $\dfrac{1 - e^{-2t}}{2}$. For $t > 1$ the overlap runs from $t - 1$ to $t$, so $$y(t) = \dfrac{e^{-2(t - 1)} - e^{-2t}}{2} = \dfrac{e^{-2t} \, (e^{2} - 1)}{2}$$

Check continuity at $t = 1$. Both formulas give $\dfrac{1 - e^{-2}}{2} = 0.432$. A mismatch there means a limit is wrong. The output rises to 0.432 at $t = 1$, then decays, and $y(2) = 0.0585$.

Where students usually get stuck

• Picking convolution limits by feel instead of sketching $x(\tau)$ and the flipped $h(t - \tau)$ for each range of $t$
• Mixing up the Fourier series coefficient formulas, or dropping the $1/T$ factor
• Skipping the region of convergence, which decides whether the same expression describes a causal or an anti-causal signal
• Treating radians per second and hertz as interchangeable, which shows up as stray factors of $2\pi$
• Mixing up the Nyquist rate (twice the highest frequency) with the Nyquist frequency (half the sampling rate)