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MSBD5004 Mathematical Methods for Data Analysis Homework 5

5004完结撒花🌸🌸ヽ(°▽°)ノ🌸🌸.

Q1

Q2

Q3

Q4

Q5

  1. Compute the Discrete Fourier Transform of [1 1 2 2]T.

Q6

Q7

Suppose length of f is N: $f=\{f[0],f[1],…,f[N-1]\}$
Then

here $W=-\frac{2\pi i}{N}$.
$\tau(f)$ is a circular right shifted signal by 1 unit. Then $\tau(f)$ is $\tau(f)=\{f[N-1], f[0], f[1], \cdots,f[N-2]\}$
The DFT of $\tau(f)$ by definition is :

Now use the property that $W^kW^{(N-1)k}=1$, and rewrite the $F(\tau(f))_k$

So the relation is $F(\tau(f))=e^{-\frac{2\pi ik}{N}}\tau(f)$.

有的地方我省略了,不懂的话私信。

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