Flashcards on LTI Systems: Frequency Domain Analysis
LTI Systems: Frequency Domain Analysis – A Student Guide
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Discrete-Time Frequency Response
10 cards
Card 1
Question: What is the discrete-time frequency response K(Ω) of an LTI system defined by impulse response g[n]
Answer: K(Ω)=∑_{n=-∞}^{∞} g[n] e^{-jΩn} (the DTFT of g[n])
Card 2
Question: How does a complex sinusoidal input x̂[n]=A e^{jΩ n} relate to the output ŷ[n] for an LTI system with frequency response K(Ω)
Answer: ŷ[n]=K(Ω) A e^{jΩ n}; the system scales the amplitude by |K(Ω)| and shifts phase by arg K(Ω) without changing frequency
Card 3
Question: What property of K(Ω) describes its behavior when Ω is increased by 2πk (integer k)
Answer: Periodicity: K(Ω+2πk)=K(Ω) for any integer k
Card 4
Question: If the impulse response g[n] is real-valued, what symmetry property does the frequency response satisfy
Answer: Conjugate symmetry: K(-Ω)=K*(Ω) (i.e., K(Ω) and K(-Ω) are complex conjugates)
Card 5
Question: Why can time-domain filtering of a finite-length or one-sided input produce a transient region different from the steady-state sinusoidal output predi
Answer: Because time-domain computation skips the 'left' part n∈(-∞,-1], causing transients until the filter reaches its AC steady state; frequency-response p
Card 6
Question: How do you obtain the frequency response from a rational transfer function G(z)=B(z)/A(z) for evaluation on the unit circle
Answer: Substitute z=e^{jΩ}: K(Ω)=G(e^{jΩ})=B(e^{jΩ})/A(e^{jΩ})
Card 7
Question: What MATLAB/Python function is commonly used to compute and plot the discrete-time frequency response?
Answer: freqz() (MATLAB) or scipy.signal.freqz (Python) to compute K(Ω) over desired frequency points and plot magnitude and phase
Card 8
Question: When using freqz with a known sampling rate Fs, how do you convert Ω to physical frequency f in Hz for plotting
Answer: f = Ω⋅Fs/(2π); plot using Omega*Fs/(2π) on the x-axis to get frequency in Hz
Card 9
Question: What two plots are typically shown to represent the frequency response and what do they display
Answer: Magnitude plot |K(Ω)| showing amplitude gain versus frequency, and phase plot arg K(Ω) showing phase shift versus frequency
Card 10
Question: Give an example numeric evaluation: if G(z)=0.2 z^{-1}/(1-0.8 z^{-1}), what is K(e^{jΩ}) at the example Ω used in the slides and the approximate magni
Answer: K(e^{jΩ})=0.2 e^{-jΩ}/(1-0.8 e^{-jΩ}); at the example Ω (used in the slides) K≈0.8186∠(-0.5377 rad) so |K|≈0.8186 and phase≈-0.5377 rad (~-31°)