10.1 Nonlinear and Time-Variant Systems
非线性与时变系统
Everything in Chapters 01-09 assumed a linear, time-invariant（LTI）system. Relaxing those two assumptions, one at a time, builds a ladder of system models. Each rung below is illustrated by one of the systems in the interactive demo that follows.
前九章都假设系统是线性时不变（LTI）的。逐一放松这两个假设，就得到一串系统模型的阶梯。下面每一级都对应后面演示中的一个系统。
LTI理想模型 · ideal model
Impulse response
depends only on the elapsed time
. Delay the input and the output simply delays with it, so no new frequency is ever created.
Demo · Linear
冲激响应只依赖经过的时间 τ；延迟输入只会让输出一起延迟，绝不产生新频率。
Interactive · shift & scale the input 互动：平移与缩放输入
The input is a narrow pulse approximating a unit impulse, so the output is essentially the impulse response itself：
. Slide
delay：y keeps its shape and only translates, tracking the input. That is
time-invariance. Slide
amplitude：y scales in proportion, shape unchanged. That is
linearity.
输入是逼近单位冲激的窄脉冲，所以输出基本上就是冲激响应本身：x(t)≈δ(t−t₀) ⟹ y(t)≈A·h(t−t₀)。拖动延迟：y 形状不变，只随输入一起平移，这就是时不变；拖动幅度：y 按比例缩放、形状不变，这就是线性。
↓时变 time-varying
LTV线性时变 · linear time-varying
Now the impulse response
also depends on the absolute time
：the same impulse is
reshaped, not merely shifted. Multiplying by a time-varying gain moves energy to the sidebands
.
Demo · Time-varying gain · Mixer
此时冲激响应 h(t,τ) 还依赖绝对时间 t：同一个冲激被改变形状，而不只是平移。乘以时变增益会把能量搬到 f±fm 的边带。
Interactive · move the impulse, watch the shape change 互动：移动冲激，观察形状变化
Here the impulse response depends on
when the impulse arrives：
changes with absolute time. At
it is exactly the LTI decaying exponential (concave)；slide
delay and its shape morphs from concave toward convex, so the output is not just a shifted copy. That is
time-variance. The
solid amber curve is the actual output；the
dashed gray curve is what a time-invariant system would give (the t₀=0 response merely shifted). Slide
amplitude：the output still scales in proportion, because an LTV system is still
linear.
此处冲激响应取决于冲激何时到达：h(t,τ) 随绝对时间变化。在 t₀=0 时它恰好就是 LTI 的衰减指数（凹）；拖动延迟，其形状由凹逐渐变凸，所以输出并非简单平移，这就是时变。实线琥珀色是实际输出；灰色虚线是若系统时不变时的输出（把 t₀=0 的响应直接平移）。拖动幅度：输出仍按比例缩放，因为 LTV 仍是线性的。
↓非线性 nonlinear
NLTV非线性时变 · nonlinear time-varying
The
Volterra series：higher-order kernels
multiply the input by itself, generating harmonics
and intermodulation
.
Demo · Square-law · Saturation · Cubic
Volterra 级数：高阶核 h2、h3… 把输入自相乘，产生 2f、3f 谐波和 f1±f2 互调。
Interactive · scale the input, watch linearity break 互动：缩放输入，观察线性失效
This system adds a Volterra second-order term：
. The
dashed gray curve is the linear prediction (the first-order part
)；the
solid red curve is the actual output. Slide
amplitude：at small A they coincide, but as A grows the actual output rises faster than linear and sharpens (the
term), so the output is no longer proportional to the input. That is
nonlinearity. Slide
delay：the shape still morphs with arrival time, so the system is also
time-variant.
该系统加入 Volterra 二阶项：y=A·h₁+βA²·h₂。灰色虚线是线性预测（一阶部分 A·h₁）；红色实线是实际输出。拖动幅度：A 较小时两者重合，A 增大时实际输出比线性增长更快、峰更尖（A²·h₂ 项），输出不再与输入成正比，这就是非线性。拖动延迟：形状仍随到达时刻变化，所以系统也是时变的。
Think About It 想一想
Q1: Why can an LTI system never create a frequency that was not in the input?为什么 LTI 系统永远无法产生输入中不存在的频率？
A complex exponential is an eigenfunction of any LTI system：feed in and the output is , the same frequency scaled by a complex number. Since any input is a sum of such exponentials and the system is linear, the output is the same set of frequencies, each multiplied by . No mechanism exists to produce a different one.
复指数是任何 LTI 系统的特征函数：输入 e^{jωn}，输出是同频率乘以复数增益 H。任意输入都是这些复指数之和，线性叠加后输出仍是同一组频率，只是各自被 H 缩放，没有任何环节能生出新频率。
Q2: Square-law on two tones gives lines at 2f1, 2f2 and f1±f2. Which are harmonics and which are intermodulation?平方律双音输出中，哪些是谐波，哪些是互调？
Squaring expands into （giving DC and the harmonics ）and the cross term （giving the intermodulation lines ）. Harmonics are integer multiples of one input tone；intermodulation lines are sums and differences of different tones, and they are the ones that fall in unexpected places.
谐波是单个输入频率的整数倍（2f1、2f2）；互调是不同频率的和差（f1±f2）。互调项往往落在意想不到的位置，是非线性失真里最棘手的部分。
Q3: Time-varying gain looks like multiplication, which is "linear in x". So why is it not an LTI system?时变增益对 x 是线性的，为什么它不是 LTI 系统？
It is linear（scaling x scales y）, but it is not time-invariant：delaying the input does not simply delay the output, because the gain is itself a function of time. Multiplying a tone by a sinusoid shifts energy to （the modulation / mixing property of the Fourier transform）. Linearity alone does not forbid new frequencies；only time-invariance does.
它对 x 是线性的（放大 x 就放大 y），但不是时不变的：增益本身随时间变化，延迟输入不等于延迟输出。正弦相乘把能量搬到 f±fm，正是傅里叶变换的调制性质。禁止产生新频率的是“时不变”，不是“线性”。