### 通过改变基向量进行探索

#### 数学设置

• 让我们从有限长度的信号开始（即$\mathbb {C} ^ {N}$中的向量）
• 傅立叶分析是对基向量的简单变换
• 基向量的改变会改观点
• 改变观点可以揭示事物（如果基向量选的好）

#### $\mathbb C^N$中的傅里叶基向量

$\left\{\mathbf{w}^{(k)}\right\}_{k=0,1}, \ldots, N-1$是$\mathbb C^N$中正交基，其中$\mathbf w_{n}^{(k)}=e^{j \frac{2 \pi}{N} n k}$，也记作$\mathbf w_k[n]$。

### DFT(离散傅里叶变换)

#### 能量分布

（注意这里基向量的模长为$N$）

### 习题

#### 1

(Difficulty: $\star$ ) Write out the phase of the complex numbers $a_{1}=1-\mathrm{j}$ and $a_{2}=-1-\mathrm{j}$
Express the phase in degrees and separate the two phases by a single white space. Each phase should be a number in the range [-180,180]

#### 2

(Difficulty: $\star$ ) Let $W_{N}^{k}=e^{-\mathrm{j} \frac{2 \pi} Nk}$ and $N>1$. Then $W_{N}^{N / 2}$ is equal to…

• -1
• 1
• $-\mathrm{j}$
• $e^{-j(2 \pi / N)+N}$

#### 3

(Difficulty: $\star$ ) Which of the following signals (continuous- and discrete-time) are periodic signals?
Note that $t \in \mathbb{R}$ and $n \in \mathbb{Z}$

• $x[n]=\sin (n)$
• $x(t)=t- \text{floor} (t)$
• $x[n]=e^{-\mathrm{j} f_{0} n}+e^{+\mathrm{j} f_{0} n},$ where $f_{0}=\sqrt{2}$
• $x(t)=\cos \left(2 \pi f_{0} t+\phi\right)$ with $f_{0} \in \mathbb{R}$
• $x[n]=1$

• $x(t)=t- \text{floor} (t)$
• $x(t)=\cos \left(2 \pi f_{0} t+\phi\right)$ with $f_{0} \in \mathbb{R}$
• $x[n]=1$

#### 4

(Difficulty: $\star \star \star)$ Choose the correct statements from the choices below.

• Consider the length- $N$ signal $x[n]=(-1)^{n}$ with $N$ even. Then $X[k]=0$ for all $k$ except $k=N / 2$
• If we apply the DFT twice to a signal $x[n],$ we obtain the signal itself scaled by $N$, i.e. $N x[n]$
• Consider the length- $N$ signal $x[n]=\cos \left(\frac{2 \pi}{N} L n+\phi\right),$ where $N$ is even and $L=N / 2 .$ Then $X[k]=\left\{\begin{array}{ll}\frac{N}{2} e^{\mathrm{j} \phi} & \text { for } k=L \ 0 & \text { otherwise }\end{array}\right.$

#### 5

(Difficulty: $\star$ ) Consider the Fourier basis $\left\{\mathbf{w}^{k}\right\}_{k=0, \ldots, N-1},$ where $\mathbf{w}^{k}[n]=e^{-j \frac{2 \pi}{N} n k}$ for $0 \leq n \leq N-1$
Select the correct statement below.

• The orthogonality of the vectors depends on the length $N$ of the elements of the basis

• The elements of the basis are orthogonal:

$\left\langle\mathbf{w}^{i}, \mathbf{w}^{j}\right\rangle=\left\{\begin{array}{ll}N & \text { for } i=j \ 0 & \text { otherwise }\end{array}\right.$

• The elements of the basis are orthonormal:
$\left\langle\mathbf{w}^{i}, \mathbf{w}^{j}\right\rangle=\left\{\begin{array}{ll}1 & \text { for } i=j \ 0 & \text { otherwise }\end{array}\right.$

#### 6

(Difficulty: $\star \star)$ Consider the three sinusoids of length $N=64$ as illustrated in the above figure; note that the signal values are shown from $n=0$ to $n=63$

Call $y_{1}[n]$ the blue signal, $y_{2}[n]$ the green and $y_{3}[n]$ the red. Further, define $x[n]=y_{1}[n]+y_{2}[n]+y_{3}[n]$
Choose the correct statements from the list below. Note that the capital letters indicate the DFT vectors.

• $Y_{1}[k]=\left\{\begin{array}{ll}N & \text { for } k=4,60 \ 0 & \text { otherwise }\end{array}\right.$
• $Y_{3}[k]=\left\{\begin{array}{ll}32 & \text { for } k=0 \ 32 & \text { for } k=64 \ 0 & \text { otherwise }\end{array}\right.$
• $Y_{2}[k]=\left\{\begin{array}{ll}16 j & \text { for } k=8 \ 16 j & \text { for } k=56 \ 0 & \text { otherwise }\end{array}\right.$
• $|x|_{2}^{2}=|X|_{2}^{2}=12800$

#### 7

(Difficulty: $\star \star \star)$ Consider the length- $N$ signal

where $M$ and $L$ are integer parameter with $0<L \leq N-1,0<M \leq N$
Choose the correct statements among the choices below.

• Consider the circularly shifted signal $y[n]=x[(n-D) \bmod N] .$ In the Fourier domain, since the DFT operator is shift invariant, it is $Y[k]=X[k]$
• In general, it will be easier to compute the norm of the signal $|\mathbf{x}|_{2}$ in the Fourier domain, using the Parseval’s Identity.
• If $M=N$ and $2 L<N$, the signal has exactly $L$ periods for $0 \leq n<N$
• For every choice of $N$ and $M$, the DFT $X[k]$ has two elements different from zero.

#### 8

(Difficulty: $\star$ ) Consider an orthogonal basis $\left\{\phi_{i}\right\}_{i=0, \ldots, N-1}$ for $\mathbb{R}^{N}$. Select the statements that hold for any
vector $\mathbf{x} \in \mathbb{R}^{N}$.

• $|\mathbf{x}|_{2}^{2}=\sum_{i=0}^{N-1}\left|\left\langle x, \phi_{i}\right\rangle\right|^{2}$
• $|\mathbf{x}|_{2}^{2}=\sum_{i=0}^{N-1}\left|\left\langle x, \phi_{i}\right\rangle\right|^{2}$ if and only if $\left|\phi_{i}\right|_{2}=1 \forall i$.
• $\begin{array}{l}|\mathbf{x}|_{2}^{2}=\frac{1}{P} \sum_{i=0}^{N-1}\left|\left\langle x, \phi_{i}\right\rangle\right|^{2} , \text { if and only if }\left|\phi_{i}\right|_{2}=P \forall i \text { . }\end{array}$
• $|\mathbf{x}|_{2}^{2}=\frac{1}{P} \sum_{i=0}^{N-1}\left|\left\langle x, \phi_{i}\right\rangle\right|^{2}$ if and only if $\left|\phi_{i}\right|_{2}^{2}=P \forall i$

#### 9

(Difficulty: $\star \star)$ Pick the correct sentence(s) among the following ones regarding the DFT $\mathbf{X}$ of a signal $\mathbf{x}$ of length $N,$ where $N$ is odd.
Remember the following definitions for an arbitrary signal (asterisk denotes conjugation):
hermitian-symmetry: $x[0]$ real and $x[n]=x[N-n]^{}$ for $n=1, \ldots, N-1$
hermitian-antisymmetry: $x[0]=0$ and $x[n]=-x[N-n]^{ }$ for $n=1, \ldots, N-1$

• If the signal $\mathbf{x}$ is hermitian-symmetric, then its DFT is real.
• If the signal $\mathbf{x}$ is purely real, then the DFT $\mathbf{X}$ is purely imaginary.
• If the signal $\mathbf{x}$ is hermitian antisymmetric, then its DFT $\mathbf{X}$ is purely imaginary.
• If the signal $\mathbf{x}$ is hermitian-symmetric, then the DFT $\mathbf{X}$ is also hermitian-symmetric.