Questions & explanations
1. Compare the impulse invariance method with the bilinear transform for designing a low-pass filter.
The impulse invariance method samples the analog impulse response, so the digital filter's time response matches the analog filter at sample times. However, it causes frequency aliasing, limiting its use to bandlimited filters. The bilinear transform uses a nonlinear mapping that compresses the entire analog frequency range into the digital range, avoiding aliasing but warping the frequency axis. For a low-pass filter, the bilinear transform gives a monotonic frequency response without aliasing, while impulse invariance may produce aliased components if the analog filter has high-frequency content. The bilinear transform is more commonly used for designing filters with sharp cutoffs, whereas impulse invariance is preferred when preserving the impulse response shape is important.
2. Compare Direct Form I and Direct Form II in terms of memory usage and numerical stability.
Direct Form I uses twice as many delay elements as Direct Form II, so it requires more memory. For an Nth-order filter, Direct Form I needs 2N delays, while Direct Form II needs N delays. However, Direct Form I is generally more numerically stable because it has separate delay lines for input and output, reducing the chance of internal overflow. Direct Form II can have large internal signals that may overflow if not scaled properly, especially for filters with high Q or sharp resonances. In fixed-point implementations, Direct Form II is more sensitive to coefficient quantization. Therefore, Direct Form I is often preferred for high-precision or critical applications, while Direct Form II is used when memory is limited.
3. Explain the read operation of a 1T DRAM cell step by step, including the role of the sense amplifier and why the read is destructive.
First, the bitline is precharged to half the supply voltage. Then the wordline activates the access transistor, connecting the capacitor to the bitline. If the capacitor stores a 1 (full voltage), charge flows from capacitor to bitline, raising its voltage slightly; if a 0 (zero voltage), charge flows from bitline to capacitor, lowering the voltage. The sense amplifier detects this small change and drives the bitline to full supply or ground. Because the capacitor loses its original charge during sharing, the read is destructive. After sensing, the sense amplifier rewrites the original value back into the cell by driving the bitline to the correct level while the wordline is still active.
4. What does the dq0 transformation do to the three-phase stator currents of an AC machine?
The dq0 transformation (also called Park transformation) takes the three stator currents (ia, ib, ic) that vary with time and rotor position, and turns them into two constant currents: one along the direct axis (d-axis) and one along the quadrature axis (q-axis), plus a zero-sequence component (0). This makes the machine equations simpler because the d and q currents are steady in steady state, like in a DC machine. The d-axis is aligned with the rotor's magnetic field, and the q-axis is perpendicular to it. The transformation uses the rotor angle to rotate the reference frame so that the d and q currents are fixed. The zero-sequence current only flows if the three phases are unbalanced.
5. Why must the substitution theorem be applied carefully when replacing a branch with a source?
The substitution theorem requires that the branch being replaced has a unique voltage and current that are consistent with the rest of the circuit. If the branch is replaced with a source that does not exactly match that voltage-current pair, the rest of the circuit will change. Also, the theorem assumes the circuit has a unique solution. In circuits with multiple solutions, like those with positive feedback, substitution might lead to an incorrect result. Additionally, the replacement must not create a conflict with Kirchhoff's laws; for example, replacing a branch with a voltage source that forces a different current than originally would violate the theorem's condition.
6. How does the impulse invariance method relate the analog and digital filter frequencies?
In the impulse invariance method, the analog frequency ω_a and digital frequency ω_d are related by ω_d = ω_a * T, where T is the sampling period. This means the digital filter's frequency response is a scaled version of the analog filter's response along the frequency axis. Because of sampling, the analog response is replicated at multiples of the sampling frequency, leading to potential aliasing. To avoid this, the analog filter must have negligible response above half the sampling rate. The mapping from the s-plane to the z-plane is given by z = e^(sT), which maps the left half of the s-plane to inside the unit circle, ensuring stability if the analog filter is stable.
7. What is the difference between Miller's theorem and the Miller effect?
Miller's theorem is a general circuit analysis tool that replaces a floating impedance between two nodes with two grounded impedances, given the voltage gain between those nodes. The Miller effect is the practical consequence of that theorem in amplifiers: the input capacitance is increased due to feedback capacitance. So, Miller's theorem is the mathematical method, while the Miller effect is the observed phenomenon of bandwidth reduction. The theorem applies to any impedance (resistor, capacitor, inductor), but the effect is most commonly discussed for capacitors because it limits high-frequency performance. Understanding both helps in designing wideband amplifiers.
8. What is the role of spectral graph theory in graph signal processing?
Spectral graph theory studies the properties of graphs through the eigenvalues and eigenvectors of matrices like the Laplacian. In GSP, it provides the foundation for the graph Fourier transform and spectral filtering. The eigenvalues indicate how signals vary across the graph: small eigenvalues correspond to low-frequency, smooth signals, while large eigenvalues correspond to high-frequency, oscillatory signals. Spectral graph theory also helps in understanding graph connectivity, clustering, and partitioning. For example, the second smallest eigenvalue (Fiedler value) relates to graph connectivity. GSP uses these spectral tools to design filters and analyze signals.
9. Compare the dq0 model of an induction machine with that of a synchronous machine: what is the main difference in the rotor?
In a synchronous machine, the rotor has a DC field winding or permanent magnets that create a fixed magnetic field, so the d-axis flux is constant. In an induction machine, the rotor has no DC source; the rotor currents are induced by slip, and the rotor flux is not fixed. So the dq0 model for an induction machine includes rotor dq windings that are shorted, and the rotor flux changes with load. Both use the same transformation to stator quantities, but the rotor equations differ. The synchronous machine has a constant field current, while the induction machine's rotor currents are zero in steady state if the rotor is open-circuited (but normally they are shorted).
10. What are the limitations of Volterra series for modeling nonlinear systems?
Volterra series have several limitations. They are best suited for weakly nonlinear systems with finite memory; for strong nonlinearities, many terms are needed, making the model impractical. The number of kernel coefficients grows exponentially with order and memory length, causing high computational cost. Volterra series cannot model systems with feedback or hysteresis, as they are essentially feedforward polynomial expansions. They also assume the nonlinearity is analytic (smooth), so they fail for discontinuous nonlinearities like hard clipping. In practice, other models like neural networks or piecewise linear approximations are used for strong nonlinearities.
11. What is the Darlington synthesis and when is it used?
Darlington synthesis realizes a positive real impedance function as a lossless two-port network terminated in a resistor. It is used when the impedance function has no real-axis zeros or poles (i.e., it is a minimum reactance function). The method involves factoring the impedance into a cascade of lossless sections (like LC ladders) and a final resistor. The resulting network uses only inductors and capacitors (no resistors except the termination), which is efficient for filter design. For example, a Butterworth low-pass filter can be synthesized using Darlington's method to produce a ladder network of L and C. This approach is common in microwave filter design.
12. When would you use fixed-point arithmetic instead of floating-point in a DSP?
Fixed-point arithmetic uses integers with a fixed number of bits for the fractional part, while floating-point uses a mantissa and exponent. Fixed-point is often used in low-cost or low-power DSPs because it requires less hardware and runs faster. However, fixed-point has limited dynamic range and can overflow if numbers get too large. Floating-point is easier to program because it handles a wide range of values automatically, but it uses more power and silicon area. You choose fixed-point when cost and speed are critical, such as in mass-market audio players, and floating-point when precision and ease of development matter, like in professional audio equipment.