Digital and analog signal processing

Trace the historical development of Fourier analysis from Fourier's heat equation work to modern signal processing applications.

Derive the Fourier series from the principle of orthogonality for complex exponentials.

Analyze the convergence of Fourier series using the Dirichlet conditions.

Prove and apply key Fourier transform properties including linearity, time-shifting, frequency-shifting, scaling, and duality.

Derive and interpret the convolution theorem in both time and frequency domains.

Apply Parseval's theorem to relate the energy of a signal in the time and frequency domains.

Derive the Fourier transforms of common signals, including rectangular pulses, sinc functions, and Gaussian functions.

Trace the development of the Laplace transform from Euler and Lagrange through Heaviside's operational calculus.

Derive the bilateral Laplace transform from the Fourier transform and define its region of convergence (ROC).

Analyze system stability and causality based on the location of poles and the ROC in the s-plane.

Apply the unilateral Laplace transform to solve linear time-invariant (LTI) differential equations with initial conditions.

Characterize analog LTI systems using their impulse response, frequency response, and transfer function.

Derive the relationship between a system's transfer function and its frequency response.

Analyze analog filter characteristics including magnitude response, phase response, and group delay.

Explain the historical development of filter approximation theory, from Butterworth to Chebyshev and Cauer (elliptic) filters.

Derive the Butterworth filter approximation and analyze its maximally flat magnitude response property.

Derive the Chebyshev Type I and Type II filter approximations and analyze their equiripple properties.

Derive the elliptic (Cauer) filter approximation and analyze its equiripple behavior in both passbands and stopbands.

Analyze the trade-offs between filter order, transition bandwidth, and passband/stopband ripple for different filter types.

Apply frequency transformations to convert a lowpass prototype filter to highpass, bandpass, and bandstop configurations.

Design an analog filter to meet given specifications by determining the required filter type, order, and pole-zero locations.

Trace the historical development of sampling theory from Nyquist's work to Shannon's formulation.

Derive the sampling theorem by analyzing the spectrum of an impulse-sampled continuous-time signal.

Analyze the phenomenon of aliasing mathematically and derive the conditions for alias-free signal reconstruction.

Derive the ideal reconstruction formula using a lowpass (sinc) interpolation filter.

Explain the practical necessity and design considerations for anti-aliasing and reconstruction filters.

Derive the relationship between continuous-time frequency (Ω) and discrete-time frequency (ω).

Derive and apply the bandpass sampling theorem for efficient sampling of band-limited signals.

Trace the historical development of the z-transform from difference equations to its modern formulation.

Derive the discrete-time Fourier transform (DTFT) and establish its relationship to the continuous-time Fourier transform of a sampled signal.

Analyze the periodicity of the DTFT spectrum and its implications.

Derive the z-transform as a generalization of the DTFT and establish its region of convergence (ROC).

Analyze the relationship between the z-plane unit circle and the DTFT frequency response.

Apply z-transform properties to analyze discrete-time LTI systems.

Analyze the stability and causality of discrete-time systems using pole-zero locations in the z-plane.

Derive the relationship between the s-plane poles of an analog system and the z-plane poles of its discrete-time counterpart via impulse invariance.

Trace the historical development of the Fast Fourier Transform (FFT) from Gauss to the Cooley-Tukey algorithm.

Derive the Discrete Fourier Transform (DFT) by sampling the DTFT spectrum.

Analyze the relationship between the DFT, DTFT, and z-transform, interpreting the DFT as samples of the z-transform on the unit circle.

Analyze spectral leakage and the picket-fence effect in DFT-based spectral analysis.

Apply various window functions (e.g., Hamming, Hanning) to reduce spectral leakage and analyze their trade-offs.

Derive the concept of circular convolution from the DFT multiplication property and distinguish it from linear convolution.

Apply zero-padding to compute linear convolution using the DFT/FFT.

Derive the decimation-in-time (DIT) FFT algorithm and analyze its computational complexity compared to the direct DFT.

Explain the historical context of digital filter design, moving from analog-to-digital transformations to direct design methods.

Design IIR digital filters using the impulse invariance method and analyze its potential for aliasing.

Design IIR digital filters using the bilinear transformation and analyze the resulting frequency warping effect.

Apply frequency pre-warping to compensate for the non-linear frequency mapping of the bilinear transform.

Design FIR filters by applying a window function to a truncated ideal impulse response.

Design FIR filters using the frequency sampling method.

Describe the principles of the Parks-McClellan algorithm for designing optimal equiripple FIR filters.

Analyze the trade-offs between IIR and FIR filters in terms of phase linearity, stability, and computational complexity.

Trace the historical development of multirate signal processing in telecommunications and audio.

Analyze the processes of decimation (downsampling) and interpolation (upsampling) in both the time and frequency domains.

Derive the polyphase representation of FIR filters for computationally efficient multirate systems.

Analyze the Short-Time Fourier Transform (STFT) for time-frequency analysis of non-stationary signals.

Apply the uncertainty principle to explain the time-frequency resolution trade-off in the STFT.

Analyze the effects of fixed-point and floating-point quantization on digital filter coefficients and signal representation.

Model quantization noise and analyze its impact on system signal-to-noise ratio (SNR).

Trace the historical development of digital audio from PCM to modern perceptual audio coding.

Analyze the characteristics of audio signals, including frequency content, dynamic range, and basic psychoacoustic properties.

Apply windowing and the STFT for the spectral analysis of audio signals.

Design and apply digital filters for audio equalization and tone control.

Analyze and implement delay-based audio effects, including simple echo and feedback-based reverberation.

Analyze and implement dynamic range compression and limiting.

Describe the constraints of real-time audio processing, including buffer size, latency, and computational load.

Set up a real-time audio input/output stream in a chosen programming environment.

Implement a windowed, overlapping buffer for processing real-time audio chunks.

Apply the FFT to buffered audio data to compute and visualize a real-time spectrogram.

Design a lowpass or bandpass filter (IIR or FIR) to meet a specific audio frequency cutoff specification.

Implement the designed filter on a real-time audio stream using its difference equation.

Design a bank of parallel bandpass filters to serve as a multi-band audio equalizer.

Implement a simple echo effect using a circular buffer for delayed signal feedback.

Develop a chorus or flanger effect by modulating the delay time of a delay line with a low-frequency oscillator.

Analyze the phase vocoder method for time-stretching and pitch-shifting.

Integrate spectral analysis, equalization, and a chosen audio effect into a single processing chain.