Digital Signal Processing

Explore spectrum representation as a powerful view of signals in the frequency domain, detailing how knowing amplitude and phase at frequencies lets you write f(t) analytically.

Derive cosine and sine formulas in terms of complex exponentials from Euler's formula, enabling analysis of a signal's spectrum.

Compute the spectrum by rewriting cosines as complex exponentials, plot lines at zero and ±5 Hz, and interpret magnitudes, phases, and beat components from sums and differences.

Derive the time-domain signal from a given spectrum by using DC components directly, and convert each spectral line into magnitude, phase, and frequency, yielding cosine terms.

Derive the spectrum of a periodic signal with Fourier series by representing it as a sum of complex exponentials and computing coefficients a_k over one period, including the dc term.

The lecture derives the Fourier series of a rectangular wave, computing the DC term and coefficients a_k, and shows how the complex-exponential series progressively approximates the signal.

Explore how discrete inputs convolve with h(n) by flipping x(n), graphically overlapping, multiplying overlapping values, and summing to produce y(n), with fir filter applications.

Explore how to determine a system's unit impulse response h[n] by feeding a unit impulse at time zero and observing the output sequence in digital signal processing.

Derive the discrete convolution difference equation for an FIR filter by flipping and sliding x(n) into h(n) and summing the products to obtain y(n).

Compare graphical and algebraic convolution approaches using h(m) and x(m), and derive the difference equation y(n)=x(n)+2x(n-1) to verify with sample inputs.

Explore delta function notations for discrete signals, derive x(n) and h(n) from delta terms, and form the convolution-based difference equation y(n) equals sum h(m) x(n-m) with a third-order filter.

Demonstrate continuous-time convolution by flipping x(tau), shifting by t, and evaluating the integral from minus infinity to infinity of x(t−tau) h(tau) d tau, yielding the x(t) * h(t) notation.

Explore convolution by visualizing the overlap of x(t) and h(t) with flipping and unit step input, yielding 1 - e^{-t} for t ≥ 0.

Explore convolving a finite pulse with a semi-infinite exponential pulse, derive piecewise output expressions for partial and full overlap, and verify symmetry by swapping x and h in the convolution.

Learn how sampling a cosine signal at delta t creates a discrete x, and how Nyquist criteria—two samples per period—prevent aliasing in the spectrum.

Define frequency response for a system with a sinusoidal input using the complex exponential form, and compute the output from the sum of h(k) e^{-j k omega_hat}.

Demonstrates that sinusoids remain sinusoids after passing through linear time-invariant systems, with outputs scaled by the frequency response magnitude and phase-shifted by its angle.

The lecture derives the frequency response for a [1,2,1] filter, expresses the magnitude as 2 cos(ω) with a phase, and computes y[n] = H(ω) x[n] at ω = π/3.

Learn how sinusoidal inputs are eigenfunctions of LTI systems; decompose into complex exponentials and apply the frequency response to scale and phase-shift each component.

Derive the frequency response and phase for simple FIR filters, showing that [1, 1] yields magnitude 2 cos(Omega/2) and phase -Omega/2, illustrating a low-pass effect; compare to [1, 2, 1].

Explore how the dtft reveals a signal's frequency content, using x(n) e^{-j omega_hat k} and h(omega_hat) = sum h(k) e^{-j omega_hat k}, and note how time discretization yields continuous frequency.

We analyze a causal, sampled sinusoid input and show how y(n) equals h(omega) x(n) in frequency-domain convolution, revealing transient and steady-state regions.

Relate the frequency response to the transfer function h(z) by defining z = r e^{j omega}. Show how h(z) extends analysis beyond unit-circle sinusoids to any complex input.

the z-transform handles non-sinusoidal signals with x(z) and h(z). convolution becomes multiplication in z-domain, with common pairs like z^{-n0} for shifted delta and 1/(1-a z^{-1}) for exponentials.

Demonstrate that convolution in the time domain equals multiplication in the z-domain using delta sequences and z-transform properties. See how the produced output matches in both domains.

learn to use the z-transform for stability analysis by identifying zeros at z = 1/3 and z = 1/2 and poles at z = 0 for a fir filter.

Use the z-transform on a two-tap system to derive both transient and steady-state responses, separating a one-step transient from the steady-state through the frequency response.

Apply z-transform techniques to analyze a discrete-time system with x[n], h[n] = delta[n] + delta[n-1], derive y[n] via z-domain multiplication, and identify transient and steady-state responses.

The lecture analyzes an IIR filter with z-transform, derives h(z), and uses partial fractions to reveal a decaying transient and a steady-state gain of four for a unit step input.