You can use SciLab (or other math software) for any part of this homework. --- ### Question 1 Suppose a signal is sampled at the rate $f_{s} = 44,100\text{ Hz}$. In each part find the smallest positive frequency which is an alias of, but not equal to, the given frequency: * **(a)** $23000\text{ Hz}$ => $21100 \pu{ Hz }$ * **(b)** $45000\text{ Hz}$ => $900\pu{ Hz }$ * **(c)** $1000\text{ Hz}$ => $43100 \pu{ Hz }$ * **(d)** $96000\text{ Hz}$ => $51900\pu{ Hz }$ --- ### Question 2 Find the signal to noise ratio $\text{SNR}$ for a signal which is sampled with 10 bit values ($B = 10$), assuming that the error is uniformly distributed between $0$ and $1/2$. $2^{10}\sqrt{ 3 }$ Now suppose a signal has values: $340, 223.45, 190.6, 48.2$, which are in the range between $-2^{9}$ and $2^{9} - 1$. Find the $\text{RMS}$ value for these four samples (take the average of the squares of the quantization errors, then take the square root). --- ### Question 3 Use a phasor sum to show that: $2 \cos(20t + \pi/3) + 3 \cos(20t + \pi/4)$ can be written as: $A \cos(20t + \phi)$. Find the constants $A$ and $\phi$ as decimal approximations. Use your project or a calculator to compute the answers and any conversions from Cartesian to polar form. Show all work for each step. --- ### Question 4 Let $B_{4} = \{v_{0}, v_{1}, v_{2}, v_{3}\}$ be the Fourier basis for dimension 4, where $v_{k}$ is the sampled phasor: $e^{i \frac{2\pi}{4}kt}, \quad t = 0, 1, 2, 3$ Let $A$ be the matrix of column vectors: $A = \begin{bmatrix} | & | & | & | \\ v_{0} & v_{1} & v_{2} & v_{3} \\ | & | & | & | \end{bmatrix}$ * **(a)** Find the determinant of $A$. * **(b)** Verify that the determinant of $A$ equals the product of backward differences from the second column, i.e., the product: $\prod_{0 \le i < j \le 3} (z_{j} - z_{i})$ where: $\begin{bmatrix} z_{0} \\ z_{1} \\ z_{2} \\ z_{3} \end{bmatrix} = \begin{bmatrix} v_{1}(0) \\ v_{1}(1) \\ v_{1}(2) \\ v_{1}(3) \end{bmatrix}$ *Note: the product notation is like a sum, where we use capital sigma ($\sum$), but now here we use capital pi ($\prod$) for product. The product is of all factors where the indexing is over all pairs $i, j$ with $i < j$ and $i, j$ in the range from $0$ to $3$.* * **(c)** Find the inverse matrix $A^{-1}$. * **(d)** Solve for the coefficients $a_{0}, a_{1}, a_{2}, a_{3}$ in the vector equation: $\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} = a_{0}v_{0} + a_{1}v_{1} + a_{2}v_{2} + a_{3}v_{3}$ using the inverse matrix. * **(e)** Solve for the coefficients $a_{0}, a_{1}, a_{2}, a_{3}$ using dot products. *(Note: for the complex dot product, however, $(cw) \cdot v = c(w \cdot v)$ and $(w \cdot cv) = \overline{c}(w \cdot v)$. So, if you multiply both sides of the equation with the dot product by $v_{i}$ on the right, then you can solve for $a_{i}$ by one division. But if you multiply with dot product on the left, then you first need to factor out $a_{i}$ from the dot product, then solve for $\overline{a_{i}}$, and finally take the conjugate to get $a_{i}$. Either way, you should get the same answer for $a_{i}$.)* * **(f)** Find the DFT of the vector: $x = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$ * **(g)** Use these coefficients to write $x$ in terms of the Fourier basis.