| 这是数字信号处理课程的一份ELEC310 Digital Signal Processing Assignment 3 代写案例Department of Electrical and Computer Engineering University of Victoria |
For the continuous-time periodic signal
$$
x(t)=2+\cos \left(\frac{2 \pi}{3} t\right)+4 \sin \left(\frac{5 \pi}{3} t\right)
$$
determine the fundamental frequency $\omega_{0}$ and the Fourier series coefficients $a_{k}$ such that
$$
x(t)=\sum_{k=-\infty}^{\infty} a_{k} e^{j k \omega_{0} t}
$$
Calculate the covariation of x(t) with sin, cos
$$
\begin{aligned}
a_{n} &=\int_{-\infty}^{\infty} x(t) \cdot \cos \left(n t\right) d t \\
b_{n} &=\int_{-\infty}^{\infty} x(t) \cdot \sin \left( n t\right) d t
\end{aligned}
$$
Use the Fourier series analysis equation to calculate the coefficients $a_{k}$ for the continuous-time periodic signal
$$
x(t)= \begin{cases}1.5, & 0 \leq t<1 \ -1.5, & 1 \leq t<2\end{cases}
$$
with fundamental frequency $\omega_{0}=\pi$.
Directly calculate the covariation of x(t) with sin, cos
$$
\begin{aligned}
a_{n} &=\int_{-\infty}^{\infty} x(t) \cdot \cos \left(n t\right) d t \\
b_{n} &=\int_{-\infty}^{\infty} x(t) \cdot \sin \left( n t\right) d t
\end{aligned}
$$
Suppose we are given the following information about a signal $x[n]$ :
- $x[n]$ is a real and even signal.
- $x[n]$ has period $N=10$ and Fourier coefficients $a_{k}$.
- $a_{11}=5$.
- $\frac{1}{10} \sum_{n=0}^{9}|x[n]|^{2}=50$.
Show that $x[n]=A \cos (B n+C)$, and specify numerical values for the constants $A$, $B$, and $C$.
Use induction on $n$
Let $x(t)$ be a periodic signal whose Fourier series coefficients are
$$
a_{k}= \begin{cases}2, & k=0 \ j\left(\frac{1}{2}\right)^{|k|}, & \text { otherwise }\end{cases}
$$
Use Fourier series properties to answer the following questions:
(a) Is $x(t)$ real?
(b) Is $x(t)$ even?
(c) Is $d x(t) / d t$ even?
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概率论代考
离散数学代写
ELEC 310 – Digital Signal Processing 1
Instructor Office Hours
F. Gebali, P.Eng., Ph.D. Everyday email or phone first
E-mail: [email protected] Location: EOW 433
Course Objectives
Generation of discrete-time signals through the sampling process and their spectral representation. Mathematical
representation and properties of digital signal processing (DSP) systems. Typical DSP systems, e.g., digital filters
and applications. The z-transform and its relation to the Laurent series. Evaluation of the inverse z transform using
complex series and contour integrals. Application of the z transform for representation and analysis of DSP systems.
The processing of continuous time signals using DSP systems. The discrete-?Fourier transform and the use of fast
Fourier transforms for its evaluation. Introduction to the design of DSP systems.
Learning Outcomes
- Understand linearity, time invariance and convolution
- Explain relation between continuous- and discrete-time Fourier transform
- Understand z-transform and its use in solving problems
- Evaluate forward and inverse z and Fourier transforms for discrete signals
- Demonstrate competency in working with both time- and frequency-domain representations of discrete-time
sampled signals - Design a discrete-time filtering algorithm based on given requirements
- Use MATLAB effectively for analysis and design of sampled digital signals
- Explain significance of sampling theorem and use it in the context of discrete-time processing of continuoustime signals
