Flashcards on Digital Signal Processing Fundamentals
Digital Signal Processing Fundamentals: A Student Guide
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Fourier analysis of signals
13 cards
Card 1
Question: What is represented by a rotating phasor in Fourier analysis of signals?
Answer: A rotating phasor represents a complex sinusoidal component; each coefficient corresponds to a rotating phasor and a pair of complex-conjugate phasors
Card 2
Question: How do two complex-conjugate phasors combine to form a real-valued sinusoid?
Answer: A complex coefficient c_k and its conjugate c_{-k} produce the real sinusoid as the sum of their rotating phasors, yielding a cosine with appropriate
Card 3
Question: What is the complex Fourier series formula for computing the coefficient c_k of a periodic signal s(t) with period T?
Answer: c_k = (1/T) ∫_{T} s(t) e^{-j (2πk/T) t} dt (equivalently c_k = (1/T) ∫_{T} s(t) e^{-j k ω_0 t} dt).
Card 4
Question: What is the inverse relation reconstructing s(t) from complex Fourier series coefficients c_k?
Answer: s(t) = Σ_k c_k e^{j (2πk/T) t} (sum over all integer k).
Card 5
Question: For a rectangular (pulse) periodic signal, how do its complex Fourier coefficients relate to the sinc function?
Answer: The coefficients c_k are proportional to sinc(k f_1 τ), where sinc(x)=sin(x)/x and f_1=1/T; specifically c_k = (pulse amplitude)·sinc(k f_1 τ) (with τ
Card 6
Question: How is the (unnormalized) sinc function defined in the slides?
Answer: sinc(x) = sin(x)/x, with sinc(0)=1 by limit.
Card 7
Question: What happens to the spectrum of a rectangular periodic signal as the period T increases while keeping pulse width fixed?
Answer: The envelope (amplitude) of discrete spectral components decreases and the spectral component density increases; in the limit T→∞ the discrete spectru
Card 8
Question: How are Fourier series coefficients normalized to transition to the Fourier transform?
Answer: Divide coefficients c_k by the frequency step Δf = 1/T and replace discrete frequencies k f_1 by continuous frequency variable f; as T→∞ (Δf→0) the di
Card 9
Question: What is the Fourier transform pair (forward and inverse) as given in the slides?
Answer: Forward: S(f) = ∫_{-∞}^{∞} s(t) e^{-j 2π f t} dt. Inverse: s(t) = ∫_{-∞}^{∞} S(f) e^{j 2π f t} df (continuous sum of rotating phasors).
Card 10
Question: How do Fourier series and Fourier transform conceptually differ according to the slides?
Answer: Fourier series expresses a periodic signal as a discrete sum of harmonics (elementary complex sinusoids); Fourier transform expresses a nonperiodic si