Chapter3 Fourier Series

Fourier Series Representation of Periodic Signals

I. Appetizer: Motivation

• Cardiovascular diseases are a significant health concern, with high mortality rates.
• Electrocardiographic (ECG) signals are a primary screening method for heart conditions.
• The lecture poses the question: Can we perform automated analysis of ECG signals to recognize different types of arrhythmias like tachycardia, bradycardia, atrial fibrillation, etc.?
• Can we differentiate arrhythmias based on automated signal analysis, for example, by looking at their periods?

II. Core Questions & Introduction to “Fourier Time”

1. Representation: Can we represent a periodic signal as a sum of periodic signal components, each with a different period?
2. Component Definition: If so, how should we define each component?
   • The answer lies in Harmonically Related Complex Exponentials (HRCE).
3. Component Strength: How do we know the strength of each component in the periodic signal?
   • This involves the definition and evaluation of Fourier series.
4. Applicability: Is this applicable to all periodic signals or just a specific class?
   • This relates to the convergence of Fourier series.

III. Complex Exponentials as the Building Block

• Why not just impulses? Convolution uses impulses to decompose signals.
• Inefficiency of Impulses: For infinitely long signals (like periodic signals), using unit impulses is inefficient.
  • $x(t) = \int_{-\infty}^{\infty} \cos(\omega_{0}\tau)\delta(t-\tau)d\tau$
• Efficiency of Complex Exponentials: Complex exponentials re themselves infinitely long, making them more efficient for representing long signals.
  • Example: $x(t) = \cos(\omega_{0}t) = \frac{1}{2}(e^{j\omega_{0}t} + e^{-j\omega_{0}t})$
• Frequency Information: Representation by complex exponentials tells us about the frequencies contained in the signal.

IV. Harmonically Related Complex Exponentials (HRCE)

• Definition: A set of signals, $\{\phi_{k}(t)|k\in\mathbb{Z}\}$, is called a set of Harmonically Related Complex Exponentials if and only if:

  $$\phi_{k}(t)=e^{jk\omega_{0}t} \text{ for each } k\in\mathbb{Z}$$

  where $\omega_{0} \ne 0$ is called the fundamental frequency.

• The frequency of $\phi_{k}(t)$ is $k\omega_{0}$ (k-fold multiple of the fundamental frequency).

• $k$ can be positive, negative, or zero (integers).

• Terminology for Harmonics:

  • $\phi_{0}(t)=1$ is called the constant term (or DC component).
  • $\phi_{\pm1}(t)$ are called the first harmonic.
  • $\phi_{\pm2}(t)$ are called the second harmonic, and so on.

• Periodicity:

  • The fundamental period is $T_{0} = \frac{2\pi}{\omega_{0}}$. So, $\phi_{k}(t)=e^{jk(\frac{2\pi}{T_{0}})t}$.
  • Although the smallest period of the k-th harmonic is $T_0/|k|$, $T_0$ is a common period for all harmonics.
  • Any linear combination $\sum_{k=-\infty}^{+\infty}a_{k}e^{jk\omega_{0}t}$ is periodic with a period of $T_{0}$.

• Adding higher-order harmonics does not change the fundamental period of the combined signal.

V. The Fourier Series Representation

• Concept: An arbitrary periodic signal may be representable by a linear combination of signals in an HRCE set that has the same fundamental frequency. This is the Fourier series representation.
• Synthesis Equation: For a periodic signal $x_{T_{0}}(t)$, its Fourier series representation is:

  $$x_{T_{0}}(t) = \sum_{k=-\infty}^{+\infty}a_{k}e^{jk\omega_{0}t}, \text{ where } \omega_{0}=2\pi/T_{0}$$

• “The theorem of Fourier series representation”: For almost every periodic signal $x_{T_{0}}(t)$, there exists a set of coefficients $\{a_{k}\in\mathbb{C}|k\in\mathbb{Z}\}$ such that the equality above holds.
• The coefficients $a_k$ need a formula for calculation.

VI. Determination of the Fourier Series Coefficients ($a_k$)

• Key Identity:

  $$\int_{T_{0}}e^{jk\omega_{0}\tau}d\tau = T_{0}\delta[k] = \begin{cases} T_{0}, & k=0 \\ 0, & k \ne 0 \end{cases}$$

  This is often written as $\int_{t}^{t+T_{0}}e^{jk\omega_{0}\tau}d\tau$.

  • Proof idea: $e^{jk\omega_{0}\tau} = \cos(k\omega_{0}\tau) + j\sin(k\omega_{0}\tau)$. The integral of a sinusoid over one full period is zero unless its frequency is zero (i.e., $k=0$).

• Analysis Equation (Formula for $a_k$):

  $$a_{k}=\frac{1}{T_{0}}\int_{T_{0}}x_{T_{0}}(t)e^{-jk\omega_{0}t}dt$$

  • Steps to calculate $a_k$:
    1. Take the periodic signal $x_{T_0}(t)$.
    2. Multiply the signal with the conjugate of the k-th order harmonic ($e^{-jk\omega_{0}t}$).
    3. Average the resultant signal over one period ($T_{0}$).

Proof.

$$\begin{aligned} \frac{1}{T_0} \int_{T_0} x_{T_0}(t) e^{-j k \omega_0 t}\, dt &= \frac{1}{T_0} \int_{T_0} \sum_{n=-\infty}^{\infty} a_n e^{j k \omega_0 t} e^{-j n \omega_0 t}\, dt \\ &= \frac{1}{T_0} \sum_{n=-\infty}^{+\infty} a_n \int_{T_0} e^{j (k-n) \omega_0 t}\, dt \\ &= \frac{1}{T_0} \sum_{n=-\infty}^{+\infty} a_n T_0 \delta[n-k] \\ &= a_k \end{aligned}$$

VII. Complete Definition of Fourier Series Representation

The Fourier series pair is given by:

1. Synthesis Equation: $x_{T_{0}}(t)=\sum_{k=-\infty}^{+\infty}a_{k}e^{jk\omega_{0}t}$
2. Analysis Equation: $a_{k}=\frac{1}{T_{0}}\int_{T_{0}}x_{T_{0}}(t)e^{-jk\omega_{0}t}dt$

• $a_k$ is called the Fourier series coefficient or the spectral coefficient of $x_{T_{0}}(t)$.
• $a[k]=\{a_{k}\}$ is called the Fourier series spectrum of $x_{T_{0}}(t)$.
• The analysis equation performs spectral analysis.

VIII. Examples of Fourier Series

1. $x_{T_{0}}(t)=C$ (a constant):
   • $a_0 = C$
   • $a_k = 0$ for $\forall k \ne 0$
   • FS representation: $x_{T_{0}}(t) = C$
2. $x_{T_{0}}(t)=\cos(\omega_{0}t)$:
   • $a_1 = \frac{1}{2}$, $a_{-1} = \frac{1}{2}$
   • $a_k = 0$ if $k \ne \pm1$
   • FS representation: $x_{T_{0}}(t) = \frac{1}{2}e^{j\omega_{0}t}+\frac{1}{2}e^{-j\omega_{0}t}$
   • Useful identity for derivation: $\int_{T_{0}}e^{j(k-n)\omega_{0}\tau}d\tau=T_{0}\delta[k-n]$

IX. Displaying the Fourier Series Spectrum

• Since $a_k$ are complex numbers, we need two plots to show the spectrum:
  1. Magnitude Spectrum: $|a_k|$ vs. $k$. Shows the strength of each harmonic.
  2. Phase Spectrum: $\angle a_k$ vs. $k$. Shows the phase (position) of each harmonic.
• We can write the synthesis equation as: $x_{T_{0}}(t) = \sum_{k=-\infty}^{+\infty}|a_{k}|e^{j(k\omega_{0}t+\angle a_{k})}$
• Example 1: $x(t)=1+\sin{\omega_{0}t}+2\cos{\omega_{0}t}+\cos(2\omega_{0}t+\frac{\pi}{4})$
  • $a_0=1$
  • $a_1=1-\frac{1}{2}j$, $a_{-1}=1+\frac{1}{2}j$ (so $|a_{\pm 1}| = \frac{\sqrt{5}}{2}$, $\angle a_1 = -\text{atan}(1/2)$, $\angle a_{-1} = \text{atan}(1/2)$)
  • $a_2=\frac{1}{2}e^{j(\pi/4)}$, $a_{-2}=\frac{1}{2}e^{-j(\pi/4)}$ (so $|a_{\pm 2}| = \frac{1}{2}$, $\angle a_2 = \pi/4$, $\angle a_{-2} = -\pi/4$)
  • $a_k=0$ for $|k|\ge3$.
• Example 2: Antisymmetric Periodic Square Wave (value +1 for $0 < t < T_0/2$, -1 for $-T_0/2 < t < 0$)
  • $a_0 = 0$
  • $a_k = \frac{1}{j\pi k}(1-(-1)^k) = \begin{cases} \frac{2}{j\pi k} & \text{for odd } k \\ 0 & \text{for even } k, k \neq 0 \end{cases}$
  • This is an odd harmonic signal (only contains odd harmonics).
• Example 3: Symmetric Periodic Square Wave (value 1 for $-T_0/4 < t < T_0/4$, 0 for $T_0/4 < |t| < T_0/2$, assuming $T_1 = T_0/4$ in the slide’s general formula context for pulse width)

• The slide shows $a_{k}=\begin{cases}1/2,&k=0\\ \frac{\sin(\pi k/2)}{\pi k}&k\ne0\end{cases}$
• This specific formula corresponds to a square wave that is 1 for half its period and 0 for the other half, centered differently or having different height/duty cycle than typical examples.
• A more standard symmetric square wave (period $T_0$) would have $a_k = \frac{\sin(k\omega_0 T_1)}{k\pi} = \frac{\sin(k\frac{2\pi}{T_0} T_1)}{k\pi}$ with $2T_1$ being the pulse width. The formula on the slide implies $T_1 = T_0/4$ and height 1.
• The phase spectrum is changed dramatically compared to the antisymmetric case due to the shift in the signal’s position.

X. Summary for Evaluating the FS Spectrum

• If the signal is sinusoidal, use Euler’s formula.

  $$\cos \varphi = \frac{e^{j\varphi} + e^{-j\varphi}}{2} \quad \sin \varphi = \frac{e^{j\varphi} - e^{-j\varphi}}{2j}$$

• If not sinusoidal, use the analysis equation:

  • $a_{0}=\frac{1}{T_{0}}\int_{T_{0}}x_{T_{0}}(t)dt$ (the average of the signal in one period).
  • $a_{k}=\frac{1}{T_{0}}\int_{T_{0}}x_{T_{0}}(t)e^{-jk\omega_{0}t}dt$.

XI. Convergence of Fourier Series

• A square wave (discontinuous) can be constructed by adding “enough” continuous complex exponential functions.

• Partial Sum: $x_{N}(t)\triangleq\sum_{k=-N}^{N}a_{k}e^{jk\omega_{0}t}$

• We want to show $\lim_{N\rightarrow+\infty}x_{N}(t)=x(t)$.

• Error Signal: $e_{N}(t)\triangleq x_{N}(t)-x(t)$

• FS representation converges if $\lim_{N\rightarrow\infty}e_{N}(t)$ is zero under a certain criterion.

• Two Convergence Criteria:

  1. Energy Convergence: $\lim_{N\rightarrow\infty}\int_{T}|e_{N}(t)|^{2}dt=0$ (the difference signal has zero energy).
  2. Pointwise Convergence: $\lim_{N\rightarrow\infty}e_{N}(t)=0$ for every $t\in\mathbb{R}$.
  • Pointwise convergence is stronger than energy convergence and implies it.

• Fourier Series Convergence Theorem #1 (Energy Convergence):

  • If a periodic signal has finite energy over a single period (i.e., $\int_{T}|x(t)|^{2}dt<\infty$), then its Fourier series partial sum converges to the signal subject to an error of zero energy.
  • The difference $x(t)-\sum_{k=-\infty}^{\infty}a_{k}e^{jk\omega_{0}t}$, if existent, must reside on a finite set of time indexes with a finite value.

• Fourier Series Convergence Theorem #2 (Pointwise Convergence):

  • If a periodic signal satisfies the Dirichlet conditions, then its Fourier series partial sum converges to the signal pointwisely, except at points of discontinuity.
  • At points of discontinuity, the Fourier series converges to the average value of the signal on either side of the discontinuity.

• Dirichlet Conditions (all three must be satisfied):

  1. $x(t)$ must be absolutely integrable over a single period: $\int_{T}|x(t)|dt<\infty$. 绝对可积
  2. $x(t)$ must have a finite number of maxima（极大值） and minima （极小值） during a single period (finite total variation). 有限个极值点，可以理解为有限次振荡
  3. $x(t)$ must have a finite number of discontinuities during a single period, and each of these discontinuities must be finite.（所有的间断点都必须是第一类间断点） 有限个第一类间断点
  • Most practical signals (e.g., speech signals, image signals) satisfy all 3 Dirichlet conditions.

XII. Gibbs Phenomenon (or Gibbs Ringing)

• When approximating a signal with discontinuities using a finite Fourier series sum ($x_N(t)$), ripples appear near the discontinuities.
• The maximal amplitude of these ripples does not decrease with increasing N. It overshoots by about 9% of the jump.
• The width of the ripples continuously decreases with increasing N.
• If $N=\infty$, the ripple width becomes 0, so effectively no ripples are present. However, for any finite N, there is always an interval near the discontinuity where ripples reside.
• This is due to the nonuniform rate of convergence of the Fourier series near discontinuities.
• Gibbs phenomenon can be observed in medical images and natural images.

傅里叶级数性质 📝

回顾：傅里叶级数的基本公式

• 合成方程 (Synthesis Equation):

  $$x_{T_{0}}(t)=\sum_{k=-\infty}^{+\infty}a_{k}e^{jk\omega_{0}t}$$

  • 表明一个周期信号可以表示为傅里叶级数，即与信号相关的谐波关系复指数 (HRCEs) 的线性组合。
• 分析方程 (Analysis Equation):

  $$a_{k}=\frac{1}{T_{0}}\int_{T_{0}}x_{T_{0}}(t)e^{-jk\omega_{0}t}dt$$

  • 用于确定傅里叶级数 (F.S.) 系数，这些系数表示每个谐波分量的强度和位置。

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吉布斯现象 (Gibbs Phenomenon)

傅里叶级数的收敛速度不同，在不连续点附近收敛较慢。

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傅里叶级数性质概览

理解傅里叶级数和傅里叶变换的性质，有助于推导信号变化时的频谱，或频谱变化时的信号。
主要性质包括：

1. 线性 (Linearity)
2. 时移 (Time shift) - 导致线性相位的增加
3. 时间反转 (Time reversal) - 导致频率反转
4. 时间尺度变换 (Time scaling) - 导致基频的改变
5. 乘法 (Multiplication) - 导致卷积
6. 微分与积分 (Differentiation & integration)
7. 共轭与对称性 (Conjugacy & Symmetry) - 对于实信号，有助于加速采集
8. 帕斯瓦尔定理 (Parseval’s theorem) - 傅里叶级数的能量守恒定理

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1. 线性 (Linearity)

• 性质: 若两个周期相同 ($T_0$) 的信号 $x(t)$ 和 $y(t)$ 对应的傅里叶级数系数分别为 $a_k$ 和 $b_k$，则对于任意常数 A 和 B：
  $Ax(t)+By(t) \stackrel{F.S.}{\longleftrightarrow} Aa_k + Bb_k$
• 证明:
  $\frac{1}{T_{0}}\int_{T_{0}}(Ax(t)+By(t))e^{-jk\omega_{0}t}dt = \frac{A}{T_{0}}\int_{T_{0}}x(t)e^{-jk\omega_{0}t}dt + \frac{B}{T_{0}}\int_{T_{0}}y(t)e^{-jk\omega_{0}t}dt = Aa_k + Bb_k$
• 意义: 两个信号线性组合的频谱是它们各自频谱的相同线性组合。

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2. 时移 (Time Shift)

• 性质: 若 $x(t) \stackrel{F.S.}{\longleftrightarrow} a_k$，则：$x(t-t_0) \stackrel{F.S.}{\longleftrightarrow} e^{-jk\omega_0 t_0} a_k$
• 意义: 时域的平移导致频域相位的线性变化 ($-k\omega_0 t_0$)。 幅度频谱不变，仅相位频谱改变。
• 证明: 设 $y(t) = x(t-t_0)$，其傅里叶系数为 $b_k$。
  $b_k = \frac{1}{T_0}\int_{T_0} y(t)e^{-jk\omega_0 t} dt = \frac{1}{T_0}\int_{T_0} x(t-t_0)e^{-jk\omega_0 t} dt$
  令 $u = t-t_0$，则 $t = u+t_0$。
  $b_k = \frac{1}{T_0}\int_{T_0} x(u)e^{-jk\omega_0 (u+t_0)} du = e^{-jk\omega_0 t_0} \left[\frac{1}{T_0}\int_{T_0} x(u)e^{-jk\omega_0 u} du\right] = e^{-jk\omega_0 t_0} a_k$

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3. 时间尺度变换 (Time Scaling)

• 性质: 时间尺度变换不改变傅里叶级数的系数/频谱。

  $$x(at) = \sum_{k=-\infty}^{+\infty} a_k e^{jk(a\omega_0)t}$$

• 注意: 尽管傅里叶级数系数 $a_k$ 不变，但谐波关系复指数集合的基频 $\omega_0$ 会变为 $a\omega_0$。
• 意义: 对信号的压缩或拉伸不改变傅里叶级数频谱本身，仅改变基频。

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4. 时间反转 (Time Reversal)

• 性质: 若 $x(t) \stackrel{F.S.}{\longleftrightarrow} a_k$，则：$x(-t) \stackrel{F.S.}{\longleftrightarrow} a_{-k}$
• 证明: 设 $y(t) = x(-t)$，其傅里叶系数为 $b_k$。
  $b_k = \frac{1}{T_0}\int_{0}^{T_0} x(-t)e^{-jk\omega_0 t} dt$
  令 $u = -t$，则 $du = -dt$。当 $t=0, u=0$; 当 $t=T_0, u=-T_0$。
  $b_k = \frac{1}{T_0}\int_{0}^{-T_0} x(u)e^{-j(-k)\omega_0 u} d(-u) = \frac{1}{T_0}\int_{-T_0}^{0} x(u)e^{-j(-k)\omega_0 u} du = a_{-k}$
• 推论:
  • 若 $x(t)$ 是偶对称 ($x(t)=x(-t)$)，则 $a_k$ 也是偶对称 ($a_k = a_{-k}$)。
  • 若 $x(t)$ 是奇对称 ($x(t)=-x(-t)$)，则 $a_k$ 也是奇对称 ($a_k = -a_{-k}$)。
• 意义: 信号反转会导致频谱也反转。

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5. 乘法 (Multiplication)

• 性质: 若两个周期相同 ($T_0$) 的信号 $x(t) \stackrel{F.S.}{\longleftrightarrow} a_k$ 和 $y(t) \stackrel{F.S.}{\longleftrightarrow} b_k$，则：

  $$x(t)y(t) \stackrel{F.S.}{\longleftrightarrow} \sum_{l=-\infty}^{\infty} a_l b_{k-l} = a_k * b_k$$

• 证明: 设 $x(t)y(t)$ 的傅里叶系数为 $c_k$。
  $c_k = \frac{1}{T_0}\int_{T_0} x(t)y(t)e^{-jk\omega_0 t} dt$
  将 $x(t) = \sum_{k_1} a_{k_1} e^{jk_1\omega_0 t}$ 和 $y(t) = \sum_{k_2} b_{k_2} e^{jk_2\omega_0 t}$ 代入：
  $c_k = \frac{1}{T_0}\int_{T_0} \left(\sum_{k_1} a_{k_1} e^{jk_1\omega_0 t}\right) \left(\sum_{k_2} b_{k_2} e^{jk_2\omega_0 t}\right) e^{-jk\omega_0 t} dt$
  $c_k = \sum_{k_1} a_{k_1} \sum_{k_2} b_{k_2} \left[\frac{1}{T_0}\int_{T_0} e^{j(k_1+k_2-k)\omega_0 t} dt\right]$
  积分项仅当 $k_1+k_2-k=0$ (即 $k_1+k_2=k$) 时为 1，否则为 0。这可以用 $\delta[k_1+k_2-k]$ 表示。
  $c_k = \sum_{k_1=-\infty}^{\infty} \sum_{k_2=-\infty}^{\infty} a_{k_1} b_{k_2} \delta[k_1+k_2-k]$
  $c_k = \sum_{k_1=-\infty}^{\infty} a_{k_1} b_{k-k_1} = a_k * b_k$
• 意义: 两个相同周期信号的乘积，其傅里叶级数频谱是两个原始频谱的卷积。

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6. 微分与积分 (Differentiation & Integration)

• 微分性质: 若 $x(t) \stackrel{F.S.}{\longleftrightarrow} a_k$，则：$\frac{dx(t)}{dt} \stackrel{F.S.}{\longleftrightarrow} jk\omega_0 a_k$
  • 可以用 $\frac{dx(t)}{dt} = \lim_{\Delta t\rightarrow0}\frac{x(t+\Delta t)-x(t)}{\Delta t}$ 来证明。
• 积分性质: 若 $x(t) \stackrel{F.S.}{\longleftrightarrow} a_k$ 且 $a_0=0$ (即信号直流分量为0)，则：$\int_{-\infty}^{t} x(\tau)d\tau \stackrel{F.S.}{\longleftrightarrow} \frac{a_k}{jk\omega_0}$ (对于 $k \neq 0$)
  • 如果 $a_0 \neq 0$，则 $\int_{-\infty}^{t} x(\tau)d\tau$ 在每个周期后会累积一个非零量，导致积分后的信号非周期性，积分性质不适用。
• 意义: 信号的微分/积分导致其频谱乘以/除以 $jk\omega_0$。
• 二阶导数频谱: $\frac{d^2x(t)}{dt^2} \stackrel{F.S.}{\longleftrightarrow} (jk\omega_0)^2 a_k = -k^2\omega_0^2 a_k$ (通过连续应用微分性质)

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7. 共轭与对称性 (Conjugacy & Symmetry)

• 共轭性质: 若 $x(t) \stackrel{F.S.}{\longleftrightarrow} a_k$，则：$x^*(t) \stackrel{F.S.}{\longleftrightarrow} a_{-k}^*$
  • 证明: 设 $x^*(t)$ 的傅里叶系数为 $b_k$。
    $b_k = \frac{1}{T_0}\int_{T_0} x^*(t)e^{-jk\omega_0 t} dt = \left(\frac{1}{T_0}\int_{T_0} x(t)e^{jk\omega_0 t} dt\right)^* = \left(\frac{1}{T_0}\int_{T_0} x(t)e^{-j(-k)\omega_0 t} dt\right)^* = a_{-k}^*$
• 重要推论:
  • 若 $x(t)$ 是实信号，即 $x(t) = x^*(t)$，则 $a_k = a_{-k}^*$。
    • 这意味着 $Re\{a_k\} = Re\{a_{-k}\}$ (实部偶对称) 且 $Im\{a_k\} = -Im\{a_{-k}\}$ (虚部奇对称)。这被称为共轭对称。
  • 若 $x(t)$ 是实且偶信号：其频谱 $a_k$ 也是实且偶的 ($Im\{a_k\}=0$, $a_k=a_{-k}$)。
    $x(t) = a_0 + 2\sum_{k=1}^{\infty} a_k \cos(k\omega_0 t)$。这表明谐波相关的余弦函数是周期实偶信号的构建模块。
  • 若 $x(t)$ 是实且奇信号：其频谱 $a_k$ 是纯虚且奇的 ($Re\{a_k\}=0$, $a_k=-a_{-k}$)。
    $x(t) = 2\sum_{k=1}^{\infty} (ja_k) \sin(k\omega_0 t)$ (注意这里的 $a_k$ 是纯虚数，所以 $ja_k$ 是实数)。这表明谐波相关的正弦函数是周期实奇信号的构建模块。
  • 基于实信号的奇偶分解，可以得出结论：几乎所有实信号都可以分解为谐波相关的正弦和余弦信号的线性组合。

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8. 帕斯瓦尔定理 (Parseval’s Theorem)

• 定理: 若 $x(t) \stackrel{F.S.}{\longleftrightarrow} a_k$，则：$\frac{1}{T_0}\int_{T_0} |x(t)|^2 dt = \sum_{k=-\infty}^{\infty} |a_k|^2$

• 意义: 周期信号在一个周期内的平均功率等于其傅里叶级数系数在整个频域上的能量

  • 或者说周期信号的总平均功率等于其所有谐波分量平均功率之和。

• 证明思路:
  $\frac{1}{T_0}\int_{T_0} |x(t)|^2 dt = \frac{1}{T_0}\int_{T_0} x(t)x^*(t) dt$
  将 $x(t) = \sum_{l} a_l e^{jl\omega_0 t}$ 和 $x^*(t) = \sum_{m} a_{-m}^* e^{-jm\omega_0 t}$ 代入，然后利用复指数的正交性，可以得到：
  $(a_k * a_{-k}^*)|_{k=0} = (\sum_{l=-\infty}^{\infty} a_l a_{-(k-l)}^*) |_{k=0} = \sum_{l=-\infty}^{\infty} a_l a_l^* = \sum_{l=-\infty}^{\infty} |a_l|^2$ (注意这里 $a_{-k}^*$ 表示 $a_k$ 的共轭反转序列，使用了傅里叶级数的共轭性质)。

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练习中的关键点

• 对于信号 $g(t)=x(at-t_0), T_0 = 4$：可以看作先进行时间平移 $x(u-t_0)$，其中 $u=at$，再进行时间尺度变换。或者，更常见的处理方式是 $g(t) = x(a(t-t_0/a))$。
  • 如果考虑 $g(t)=x(2t-1)$：
    • 设 $y(t) = x(t-1)$，则 $Y_k = a_k e^{-jk\omega_0} = a_k e^{-jk\frac{\pi}{2}}$
    • 然后 $g(t) = y(2t)$。时间尺度变换不改变系数，但基频变为 $2\omega_0$。所以 $g(t)$ 的系数仍然是 $a_k e^{-jk\frac{\pi}{2}}$，但其展开式为 $\sum a_k e^{-jk\frac{\pi}{2}} e^{jk(2\omega_0)t}$。
  • 对于 $g(t)=x(-2t-1)$：
    • 先令 $y(t)=x(t-1)$, $Y_k = a_k e^{-jk\omega_0} = a_k e^{-jk\frac{\pi}{2}}$。
    • 再令 $z(t)=y(-t)$, $Z_k = Y_{-k} = a_{-k}e^{jk\frac{\pi}{2}}$。
    • 最后 $g(t)=z(2t)$, $G_k = Z_k = a_{-k}e^{jk\frac{\pi}{2}}$，基频变为 $\omega = 2\omega_0 = \pi$。

• 一个奇对称的实信号 $x(t)$ (如图)，其傅里叶系数 $a_k$ 是纯虚且奇对称的。
• 如果 $x(t)$ 的傅里叶系数是 $a_k$，则 $x'(t)$ 的傅里叶系数 $b_k = jk\omega_0 a_k$。如果 $a_k$ 是纯虚且奇对称，则 $jk\omega_0 a_k$ 将会是实且偶对称的。

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Bonus 问题解答

• 问题: 假设要重建的图像是实值信号。如果只采集了一半的频谱，如何无损重建图像？直接使用综合方程会产生错误，如何无损重建？
• 简要回答:
  由于图像是实值信号，其傅里叶频谱具有共轭对称性 ($A(k_x, k_y) = A^*(-k_x, -k_y)$)。 如果只采集了一半的频谱（例如，对于正频率部分），另一半频谱可以通过共轭对称性推算出来。获得完整的频谱信息后，就可以通过傅里叶逆变换（对于图像是二维傅里叶逆变换，类似综合方程）无损重建图像。直接使用不完整的（只有一半的）频谱进行逆变换当然会出错。

I. Eigenfunction Properties of LTI Systems

A. Definition

• An eigenfunction (特征函数) of an LTI system is an input signal that produces an output signal that is simply the input multiplied by a constant.
• If $x(t)*h(t)=\lambda x(t)$, then $x(t)$ is an eigenfunction of the system $h(t)$, and $\lambda$ is the eigenvalue (特征值) associated with it.

B. Notes on Eigenfunctions

• For an eigenfunction input, predicting the output is a simple scaling operation, which is much easier than convolution.
• Most input signals are NOT eigenfunctions.
• Different LTI systems generally have different eigenfunctions.

C. Complex Exponentials as Eigenfunctions

A crucial property is that complex exponentials are ALWAYS eigenfunctions of any LTI system.

Proof:

• Let the input be $x(t)=e^{j\omega t}$. The output $y(t)$ is given by the convolution integral:

  $$y(t) = \int_{-\infty}^{\infty}x(t-\tau)h(\tau)d\tau = \int_{-\infty}^{\infty}e^{j\omega(t-\tau)}h(\tau)d\tau = e^{j\omega t}\int_{-\infty}^{\infty}e^{-j\omega\tau}h(\tau)d\tau$$

• Since the integral $\int_{-\infty}^{\infty}e^{-j\omega\tau}h(\tau)d\tau$ does not depend on $t$, we can define it as $H(j\omega)$.

• So, the output is $y(t) = H(j\omega)e^{j\omega t}$.

• Here, $H(j\omega) \triangleq \int_{-\infty}^{\infty}e^{-j\omega\tau}h(\tau)d\tau$ is the eigenvalue associated with the eigenfunction $e^{j\omega t}$.

• This eigenvalue, $H(j\omega)$, is known as the frequency response of the LTI system and depends on frequency $\omega$ and the system’s impulse response $h(\tau)$, but not on time $t$.

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II. Frequency Domain Analysis of LTI Systems

A. Response to Periodic Signals

• If a periodic input signal $x_{T_0}(t)$ is represented by its Fourier series:

$$x_{T_{0}}(t)=\sum_{k=-\infty}^{+\infty}a_{k}e^{jk\omega_{0}t}$$

where $e^{jk\omega_{0}t}$ are eigenfunctions of the LTI system.

• The response of the LTI system $T\{\cdot\}$ to this arbitrary periodic signal is:

$$y(t) = T\{x_{T_0}(t)\} = T\{\sum_{k=-\infty}^{\infty}a_{k}e^{jk\omega_{0}t}\}$$

• Using linearity and the eigenfunction property $T\{e^{jk\omega_{0}t}\} = H(jk\omega_0)e^{jk\omega_{0}t}$:

  $$y(t) = \sum_{k=-\infty}^{\infty}a_{k}T\{e^{jk\omega_{0}t}\} = \sum_{k=-\infty}^{\infty}a_{k}[H(jk\omega_{0})e^{jk\omega_{0}t}] = \sum_{k=-\infty}^{\infty}[a_{k}H(jk\omega_{0})]e^{jk\omega_{0}t}$$

• If we let $b_k = a_k H(jk\omega_0)$, then the output is also a periodic signal with the same period:

  $$y(t) = \sum_{k=-\infty}^{\infty}b_{k}e^{jk\omega_{0}t}$$

• The output spectrum is $b_k = a_k H(jk\omega_0)$.

• $H(jk\omega_0)$ describes how the system modifies each harmonic component of the input signal. It’s a sampled version of the continuous frequency response $H(j\omega)$ at discrete harmonic frequencies $\omega = k\omega_0$.

B. Two Methods to Analyze an LTI System

1. Time Domain:
   • Input: $x(t)=\int_{-\infty}^{\infty}x(\tau)\delta(t-\tau)d\tau$ (sum of impulses)
   • System: Impulse response $h(t)$
   • Output: $y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$ (convolution)
2. Frequency Domain (for periodic inputs):
   • Input: $x(t)=\sum_{k=-\infty}^{\infty}a_{k}e^{jk\omega_{0}t}$ (sum of exponentials)
   • System: Frequency response $H(j\omega)=\int_{-\infty}^{\infty}e^{-j\omega\tau}h(\tau)d\tau$, sampled at $k\omega_0 \Rightarrow H(jk\omega_0)$
   • Output: $y(t)=\sum_{k=-\infty}^{\infty}(a_{k}H(jk\omega_{0}))e^{jk\omega_{0}t}$ (sum of exponential responses)

C. Rationale for Frequency-Domain Analysis

1. Computational Advantages: Multiplication in the frequency domain is often simpler than convolution in the time domain.
   • Example: Input $x(t)=\cos(\frac{2}{3}t)$ and impulse response $h(t)=\frac{\sin(t)}{\pi t}$.
     • The frequency response is $H(j\omega)=\int_{-\infty}^{\infty}e^{-j\omega\tau}\frac{\sin(\tau)}{\pi\tau}d\tau = \begin{cases}1, & \text{if } |\omega|<1 \\ 0, & \text{otherwise}\end{cases}$ (This is a standard Fourier Transform pair for an ideal low-pass filter scaled).
     • For $x(t)=\cos(\frac{2}{3}t)$, $\omega_0 = \frac{2}{3}$. The Fourier series coefficients $a_k$ are non-zero only for $k=-1$ and $k=1$ (specifically, $a_1=a_{-1}=1/2$).
     • The relevant system responses are at $H(j1\cdot\frac{2}{3}) = H(j\frac{2}{3})$ and $H(j(-1)\cdot\frac{2}{3}) = H(-j\frac{2}{3})$.
     • Since $|\pm \frac{2}{3}| < 1$, $H(j\frac{2}{3})=1$ and $H(-j\frac{2}{3})=1$.
     • The output spectrum $b_k = a_k H(jk\omega_0)$ will be $b_1 = a_1 \cdot 1 = 1/2$ and $b_{-1} = a_{-1} \cdot 1 = 1/2$.
     • Thus, $y(t)=x(t)$.
2. Critical Insights: It shows how the system modifies each harmonic.
   • If $|H(jk\omega_{0})| > 1$, the $k^{th}$ harmonic is strengthened.
   • If $|H(jk\omega_{0})| < 1$, the $k^{th}$ harmonic is attenuated.
   • This leads to the concept of filtering.

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III. Filtering (滤波)

A. Definition

Filtering is essentially spectral editing. An LTI filter is an LTI system whose frequency response $H(j\omega)$ is shaped to change the input spectrum by multiplication ($b_k = a_k H(jk\omega_0)$).

• Example: Lowpass Filter (低通滤波器)
  • $H(j\omega) = 1$ for $|\omega| < \omega_c$ (Passband)
  • $H(j\omega) = 0$ for $|\omega| > \omega_c$ (Stopband)
  • For harmonics $k\omega_0$ within $(-\omega_c, \omega_c)$, $a_k$ is preserved ($b_k=a_k$).
  • For harmonics outside this range, $a_k$ is suppressed ($b_k=0$).
  • Low-frequency harmonics are preserved; high-frequency ones are eliminated.

B. Types of LTI Filters

1. Frequency-selective filters (频率选择性滤波器) or ideal filters (理想滤波器): Pass certain frequencies and completely eliminate others.
   • Ideal Lowpass Filter (低通): Kills high frequencies. $H(j\omega)=1$ for $|\omega|<\omega_c$, 0 otherwise.
   • Ideal Highpass Filter (高通): Kills low frequencies. $H(j\omega)=1$ for $|\omega|>\omega_c$, 0 otherwise. (The image shows passband outside $(-\omega_c, \omega_c)$ but it should typically be symmetric, passing $|\omega| > \omega_c$).
   • Ideal Bandpass Filter (带通): Passes a specific band of frequencies. $H(j\omega)=1$ for $\omega_{c1}<|\omega|<\omega_{c2}$, 0 otherwise.
2. Frequency-shaping filters (频率成形滤波器): Change the shape of the input spectrum without completely passing or killing harmonics.
   • Example: Differentiator ($y(t) = \frac{dx(t)}{dt}$).
     • Its frequency response is $H(j\omega) = j\omega$.
     • Magnitude: $|H(j\omega)| = |\omega|$.
     • High-frequency components are strengthened, and low-frequency components are suppressed.

C. Filter Questions

• Is an edge detector a low-pass or high-pass filter?
  • Edges are determined by high-frequency harmonics. Since edge detectors strengthen these, they are high-pass filters.
• What filter to use to keep only harmonics within 10-500Hz for an EMG signal?
  • This requires a bandpass filter because it preserves a specific band of frequencies, eliminating those below 10Hz and above 500Hz.

Looking at this table of properties for continuous-time Fourier series, I’ll convert it to markdown format with proper LaTeX formula syntax:

Property	Section	Periodic Signal	Fourier Series Coefficients
		$x(t)$ Periodic with period $T$ and $y(t)$ fundamental frequency $\omega_0 = 2\pi/T$	$a_k$ $b_k$
Linearity	3.5.1	$Ax(t) + By(t)$	$Aa_k + Bb_k$
Time Shifting	3.5.2	$x(t - t_0)$	$a_k e^{-jk\omega_0 t_0} = a_k e^{-jk(2\pi/T)t_0}$
Frequency Shifting		$e^{jM\omega_0 t} (= e^{jM(2\pi/T)t})x(t)$	$a_{k-M}$
Conjugation	3.5.6	$x^*(t)$	$a^*_{-k}$
Time Reversal	3.5.3	$x(-t)$	$a_{-k}$
Time Scaling	3.5.4	$x(\alpha t), \alpha > 0$ (periodic with period $T/\alpha$)	$a_k$
Periodic Convolution		$\int_T x(\tau)y(t - \tau)d\tau$	$Ta_k b_k$
Multiplication	3.5.5	$x(t)y(t)$	$a_k * b_k = \sum_{l=-\infty}^{\infty} a_l b_{k-l}$
Differentiation		$\frac{dx(t)}{dt}$	$jk\omega_0 a_k = jk\frac{2\pi}{T}a_k$
Integration		$\int_{-\infty}^t x(t)dt$ (finite valued and periodic only if $a_0 = 0$)	$\left(\frac{1}{jk\omega_0}\right)a_k = \left(\frac{1}{jk(2\pi/T)}\right)a_k$
Conjugate Symmetry for Real Signals	3.5.6	$x(t)$ real	$a_k = a^*_{-k}$ $\mathfrak{Re}\{a_k\} = \mathfrak{Re}\{a^*_{-k}\}$ $\mathfrak{Im}\{a_k\} = -\mathfrak{Im}\{a_{-k}\}$ $
Real and Even Signals	3.5.6	$x(t)$ real and even	$a_k$ real and even
Real and Odd Signals	3.5.6	$x(t)$ real and odd	$a_k$ purely imaginary and odd
Even-Odd Decomposition of Real Signals		$x_e(t) = \mathcal{E}v\{x(t)\}$ [$x(t)$ real] $x_o(t) = \mathcal{O}d\{x(t)\}$ [$x(t)$ real]	$\mathfrak{Re}\{a_k\}$ $j\mathfrak{Im}\{a_k\}$

Parseval’s Relation for Periodic Signals

$$\frac{1}{T}\int_T |x(t)|^2 dt = \sum_{k=-\infty}^{\infty} |a_k|^2$$