傅里叶分析与谱估计直观指南(1-4章)

傅立叶变换

频谱估计

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"The Intuitive Guide to Fourier Analysis and Spectral Estimation(1-4章) - Charan Langton & Victor Levin" 这本教材《直观指南：傅立叶分析与频谱估计》由Charan Langton和Victor Levin共同撰写，旨在以通俗易懂的方式介绍傅立叶变换和频谱估计的基本概念。尽管章节可能不完整（只包含1-4章），但它仍然是学习这两个关键领域的宝贵资源。傅立叶变换是数学中的一个重要工具，它将一个函数或信号从时域表示转换到频域表示。在工程、物理、信号处理等领域，傅立叶变换的应用广泛。它可以帮助我们理解信号的频率成分，揭示隐藏的周期性和模式。书中可能介绍了基本的连续傅立叶变换、离散傅立叶变换（DFT）以及快速傅立叶变换（FFT）的概念，这些都是实际应用中计算傅立叶变换的常用方法。频谱估计是信号处理中的一个核心主题，它涉及到识别和量化信号中的频率成分。通过对信号进行傅立叶分析，我们可以估计其功率谱，即信号在不同频率上的功率分布。这对于理解和解析复杂信号，如通信信号、音频信号或医学图像等，至关重要。书中可能涵盖了不同的频谱估计方法，如窗口函数、平均谱估计、最大似然估计等，并讨论了它们的优缺点和适用场景。教材的前四章可能会涵盖以下内容： 1. 傅立叶变换的基础：解释傅立叶变换的基本原理，如何通过积分或离散化计算来实现，以及它的物理意义。 2. 离散傅立叶变换和快速傅立叶变换：介绍DFT和FFT的定义、计算过程，以及它们在数字信号处理中的应用。 3. 傅立叶变换的性质：探讨傅立叶变换的线性性质、共轭对称性、平移和尺度变换等特性，以及这些性质如何帮助我们理解和操作信号。 4. 频谱估计基础：阐述频谱估计的基本概念，介绍几种基本的谱估计方法，并通过实例展示如何进行频谱分析。教材还提供了MATLAB代码和其他补充材料的网站，这有助于读者更深入地理解和应用所学知识。通过实践编程，读者可以亲手计算和可视化傅立叶变换和频谱估计的结果，从而增强理解和掌握。《直观指南：傅立叶分析与频谱估计》提供了一个友好的学习平台，让初学者能够轻松入门这两个复杂但重要的领域。通过阅读和实践，读者可以建立起对傅立叶变换和频谱估计的直观理解，为未来在相关领域的研究或工作打下坚实基础。

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CHAPTER 1. TRIGONOMETRIC REPRESENTATION OF CT PERIODIC SIGNALS

Frequency and Time Domain Views of a Signal

Consider the wave in Fig. 1.2. One would be hard pressed to guess its equation. Yet, it is just

a sum of three waves as shown in Fig. 1.3(a), 1.3(b) and 1.3(c) of differing frequencies and

amplitudes.

0 0.5 1 1.5 2 2.5 3 3.5 4

−4

−2

Time

Amplitude

Figure 1.2: An arbitrary periodic wave for which we would like to know its equation.

0 2 4

−2

Amplitude

(a) Wave 1

0 2 4

−2

Time

(b) Wave 2

0 2 4

−2

Figure 1.3: The components of the arbitrary wave of ﬁgure 1.2.

Therefore, although we know that the wave of Fig. 1.2 is created from the sum of three

regular-looking sinusoids of Fig. 1.3, how could we ﬁgure this out, if we did not already

know the answer?

Spectrum of a signal

Let us look at Fig. 1.4 which shows a general signal in three dimensions. When we see a

signal in time, we are looking at it in what is called, the Time Domain. What we are actually

observing is a composite signal. It is a sum of components which we are unable to see. In

this three-dimensional view, when we look at a signal from the side view, each component

appears as a single vertical line at its own discrete frequency. This side view of the signal is

called the Frequency Domain. Another name for this view is the Signal Spectrum.

The spectrum is a way to quantify the component frequencies. By quantify we mean identifying

the amplitude of each of the components. The spectrum of the signal shown in Fig. 1.2 is

composed of just three frequencies and can be drawn as illustrated in Fig. 1.5(a). This is

called a one-sided amplitude spectrum. The x-axis in Fig. 1.5(a) represents the component

frequencies of the signal, whereas the y-axis is the amplitude of those frequencies. Fig. 1.5(a)

CHAPTER 1. TRIGONOMETRIC REPRESENTATION OF CT PERIODIC SIGNALS

A set of waves is harmonic if its frequency is an integer multiple of the fundamental wave’s

frequency. We can write this as a set.

c(t) = cos(2πf

s(t) = sin(2π f

(1.4)

We have introduced an index, k in Eq. (1.4) such that each harmonic frequency is equal to k

times the fundamental frequency, f

= k f

, with k any arbitrary integer, 0, 1, 2, . . . , ∞.

In Fourier series formulation, the index k spans all positive integers to inﬁnity, including zero.

Note that the fundamental wave or the ﬁrst harmonic is often deﬁned as the one for k = 1.

Hence, the frequency of the ﬁrst harmonic and the fundamental frequency are the same. The

wave obtained for k = 0 is of course nothing but a ﬂat line. The wave for k = 2 is called the

second harmonic and so on for higher values of the index k.

Let us rewrite the deﬁnition of the harmonics allowing the phase and the amplitude to vary.

We give each harmonic a unique amplitude and phase and rewrite the harmonic signals as:

(t) = a

cos(2πkf

t + φ

)

(t) = b

sin(2πkf

t + φ

)

(1.5)

The wave c

(t) is a cosine wave of kth harmonic frequency or k f

, its amplitude being a

and the phase in radians being φ

. The signal s

(t) is a sine wave with similarly unique

amplitude and phase for the same harmonic frequency, k f

. Now although the frequencies

are still related by multiples of integer k, we are allowing the amplitude and the phase of

each harmonic to be different. Such waves are still considered harmonic. The amplitude

coefﬁcient a

for a cosine and b

for a sine are now arbitrary.

What is amplitude? Amplitude is often thought of as the value of a wave’s height above

its mean value at any particular time. The amplitude can be either positive or negative,

indicating where it is being measured, above or below the mean value. If a sinusoid is given

by the expression: a cos(ωt), then the instantaneous value of the sinusoid (±1) is scaled by

the coefﬁcient, a. The peak value is never more than a, hence this coefﬁcient is the peak

amplitude. This coefﬁcient, is called the amplitude of the wave. It is one half of the wave’s

full excursion above and below its mean. In Eq. (1.5), the terms a

and b

are the amplitudes

of the waves.

What is phase? The argument of a sinusoid sin(θ ), is in fact an angle, or the term θ .

However, in signal processing, we often need to represent a sinusoid as a function of time.

We do this by writing the sinusoid as sin(ωt), where ω is the radial frequency deﬁned in

terms of angles per time. Both forms of the sinusoid argument, the θ and its equivalent form

CHAPTER 1. TRIGONOMETRIC REPRESENTATION OF CT PERIODIC SIGNALS

ωt, are called the instantaneous phase of the sinusoid. We can shift a sinusoid, or in fact

any wave in time, by adding a term, φ, to the instantaneous phase, θ , writing the sinusoid

as sin(θ + φ). This second term, φ, is assumed to be ﬁxed as a function of time (for linear

systems) and is commonly called the phase. This can be confusing, because we have two

quantities here, both called phase. The total phase at any time is made up of these two parts,

one a ﬁxed quantity and the other changing with time. However, generally when we say

phase, we are referring not to the instantaneous quantity ωt, but to the ﬁxed quantity, φ.

This term is more properly called the phase shift but the qualiﬁer word shift is often left off.

In Eq. (1.5), the term φ

is the phase (shift) for the kth wave.

Figure 1.7(a) shows that the amplitude of the sinusoid is 1.0, even though the full excursion

is 2.0. Figure 1.7(b) shows that a phase shift of φ = π/2 has turned the sine wave into a

cosine wave. This phase shift has no effect on the amplitude or the frequency of the wave,

hence phase, amplitude and frequency are independent terms.

-1

Amplitude

(a) Sine

-1

(b) Sine with phase shift

θθ

sin( )θ

sin( )

θ +

Figure 1.7: The phase of a sinusoid consists of its instantaneous and changing phase, (θ = ωt), plus a

constant phase term, φ, called the phase shift as shown in (b).

The sine and cosine waves differ in phase by π/2 radians, or 90

◦

. When two signals differ

in phase by −90

◦

or +90

◦

, they are said to be in quadrature, hence sine and cosine waves

are in quadrature. For linear Fourier analysis, it is assumed that phase (the φ term) and

frequency are not a function of time, i.e., both the phase and the frequency of a signal do

not change over time. This is one of the fundamental assumptions of Fourier analysis. All

signals subjected to Fourier analysis are assumed to have ﬁxed and unchanging frequencies

as well as the phase term, φ. This assumption implies that the signal is stationary, a concept

discussed in Chapter 8.

In Fig. 1.8, two harmonics of a sinusoid of frequency 1 Hz are seen, with sines on the left side

and cosines on the right. Both the sines and cosines have a peak amplitude of 1. Depending

on your ﬁeld of interest, the units of amplitude can be pretty much anything. In this book,

the signal amplitude units will be in volts and phase in radians.

The cosine always reaches its peak amplitude at time t = 0, and has a phase of π/2 at

that time. The sine, no matter what the frequency, always has an amplitude of zero at

time t = 0, which is equivalent to a phase of 0 radians. Hence, no matter how many sine

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