Flashcards on Z-Transforms and Transfer Functions in DSP
Z-Transforms and Transfer Functions in DSP: The Ultimate Guide
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Discrete-time LTI Systems and Z-transform
14 cards
Card 1
Question: What is the system transfer function G(z) of a discrete-time LTI system in terms of the Z-transform of its impulse response g[n] and input X(z)?
Answer: G(z) = Z{g[n]} and Y(z) = G(z) · X(z) where Y(z) is the Z-transform of the output y[n].
Card 2
Question: How does linearity appear in the operator (Z) domain for an LTI system with input ax[n]?
Answer: If y[n]=a x[n], then Y(z)=a X(z); linear scaling in time domain becomes scaling of the Z-transform by a constant in the operator domain.
Card 3
Question: How does addition of signals in time domain appear in the Z domain?
Answer: If y[n]=x1[n]+x2[n], then Y(z)=X1(z)+X2(z); addition maps to addition of Z-transforms.
Card 4
Question: What is the Z-domain effect of a one-sample delay z^{-1} on x[n]?
Answer: If y[n]=x[n-1], then Y(z)=z^{-1} X(z). Delay in time corresponds to multiplication by z^{-1} in the Z domain.
Card 5
Question: How can every LTI system transfer function be expressed algebraically?
Answer: As a rational polynomial G(z)= (B0 + B1 z^{-1} + ... + BN z^{-N}) / (A0 + A1 z^{-1} + ... + AN z^{-N}).
Card 6
Question: What normalization can be done to the rational polynomial transfer function when A0 ≠ 0?
Answer: Divide numerator and denominator by A0 to obtain a normalized form with leading denominator coefficient 1 (i.e., G(z) with denominator 1 + a1 z^{-1} +
Card 7
Question: What are Direct Form I and Direct Form II implementations of a transfer function?
Answer: They are canonical signal-flow realizations of the rational transfer function. Direct Form I implements numerator and denominator sections with separa
Card 8
Question: Why might Direct Form II be less suitable for fixed-point implementations?
Answer: Direct Form II uses fewer delay blocks but is more sensitive to overflow, making it less suitable for fixed-point arithmetic.
Card 9
Question: Given G(z) = (1 + 2 z^{-1})/(1 - 0.9 z^{-1}), which direct form was used in the example to draw the signal flow?
Answer: Direct Form I (Canonical Form I) was used in the example.
Card 10
Question: How can MATLAB or Python compute the output of an LTI system given its transfer function coefficients and the whole input signal x[n]?
Answer: Use filter(b,a,x) in MATLAB or lfilter(b,a,x) in Python (scipy.signal), where b = [B0,B1,...,BN] and a = [A0,A1,...,AN].