Add questions for pset 12

_psets/12.md

+---
+title: Problem Set 12
+---
+
+## (15.1)
+
+### (a)
+
+{:.question}
+Show that the circuits in Figures 15.1 and 15.2 differentiate, integrate, sum, and difference.
+
+### (b)
+
+{:.question}
+Design a non-inverting op-amp amplifier. Why are they used less commonly than inverting ones?
+
+### (c)
+
+{:.question}
+Design a transimpedance (voltage out proportional to current in) and a transconductance (current out
+proportional to voltage in) op-amp circuit.
+
+### (d)
+
+{:.question}
+Derive equation (15.16).
+
+## (15.2)
+
+{:.question}
+If an op-amp with a gain–bandwidth product of 10 MHz and an open-loop DC gain of 100 dB is
+configured as an inverting amplifier, plot the magnitude and phase of the gain as a function of
+frequency as $$R_\text{out}/R_\text{in}$$ is varied.
+
+## (15.3)
+
+{:.question}
+A lock-in has an oscillator frequency of 100 kHz, a bandpass filter Q of 50 (re- member that the Q
+or quality factor is the ratio of the center frequency to the width between the frequencies at which
+the power is reduced by a factor of 2), an input detector that has a flat response up to 1 MHz, and
+an output filter time constant of 1 s. For simplicity, assume that both filters are flat in their
+passbands and have sharp cutoffs. Estimate the amount of noise reduction at each stage for a signal
+corrupted by additive uncorrelated white noise.
+
+## (15.4)
+
+### (a)
+
+{:.question}
+For an order 4 maximal LFSR work out the bit sequence.
+
+### (b)
+
+{:.question}
+If an LFSR has a chip rate of 1GHz, how long must it be for the time between repeats to be the age
+of the universe?
+
+### (c)
+
+{:.question}
+Assuming a flat noise power spectrum, what is the coding gain if the entire sequence is used to send
+one bit?
+
+## (15.5)
+
+{:.question}
+What is the SNR due to quantization noise in an 8-bit A/D? 16-bit? How much must the former be
+averaged to match the latter?
+
+## (15.6)
+
+{:.question}
+The message 00 10 01 11 00 ($$c_1$$, $$c_2$$) was received from a noisy channel. If it was sent by
+the convolutional encoder in Figure 15.20, what data were transmitted?
+
+## (15.7)
+
+{:.question}
+This problem is harder than the others.
+
+### (a)
+
+{:.question}
+Generate and plot a periodically sampled time series {$$t_j$$} of N points for the sum of two sine
+waves at 697 and 1209 Hz, which is the DTMF tone for the number 1 key.
+
+### (b)
+
+{:.question}
+Calculate and plot the Discrete Cosine Transform (DCT) coefficients {$$f_i$$} for these data,
+defined by their multiplication by the matrix $$f_i = \sum_{j = 0}^{N - 1} D_{ij} t_j$$, where
+
+$$
+\begin{align*}
+D_{ij} =
+\begin{cases}
+\sqrt{\frac{1}{N}} &(i = 0)\\
+\sqrt{\frac{2}{N}} \cos \left( \frac{\pi (2j + 1) i}{2 N} \right) &(1 \leq i \leq N - 1)
+\end{cases}
+\end{align*}
+$$
+
+### (c)
+
+{:.question}
+Plot the inverse transform of the {$$f_i$$} by multiplying them by the inverse of the DCT matrix
+(which is equal to its transpose) and verify that it matches the time series.
+
+### (d)
+
+{:.question}
+Randomly sample and plot a subset of M points {$$t^\prime_k$$} of the {$$t_j$$}; you’ll later
+investigate the dependence on the sample size.
+
+### (e)
+
+{:.question}
+Starting with a random guess for the DCT coefficients {$$f^\prime_i$$}, use gradient descent to
+minimize the error at the sample points
+
+$$
+\min_{\{f^\prime_i\}} \sum_{k = 0}^{M - 1}
+\left( t^\prime_k - \sum_{j = 0}^{N - 1} D_{ij} f^\prime_i \right)^2
+$$
+
+{:.question}
+and plot the resulting estimated coefficients.
+
+### (f)
+
+{:.question}
+The preceding minimization is under-constrained; it becomes well-posed if a norm of the DCT
+coefficients is minimized subject to a constraint of agreeing with the sampled points. One of the
+simplest (but not best [Gershenfeld, 1999]) ways to do this is by adding a penalty term to the
+minimization. Repeat the gradient descent minimization using the L2 norm:
+
+$$
+\min_{\{f^\prime_i\}} \sum_{k = 0}^{M - 1}
+\left( t^\prime_k - \sum_{j = 0}^{N - 1} D_{ij} f^\prime_i \right)^2
++ \sum_{i = 0}^{N - 1} f^{\prime 2}_i
+$$
+
+{:.question}
+and plot the resulting estimated coefficients.
+
+### (g)
+
+{:.question}
+Repeat the gradient descent minimization using the L1 norm:
+
+$$
+\min_{\{f^\prime_i\}} \sum_{k = 0}^{M - 1}
+\left( t^\prime_k - \sum_{j = 0}^{N - 1} D_{ij} f^\prime_i \right)^2
++ \sum_{i = 0}^{N - 1} \vert f^{\prime}_i \vert
+$$
+
+{:.question}
+Plot the resulting estimated coefficients, compare to the L2 norm estimate, and compare the
+dependence of the results on M to the Nyquist sampling limit of twice the highest frequency.