Python实践：Think DSP——探索数字信号处理

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4 Chapter 1. Sounds and signals

0 2000 4000 6000 8000 10000

Frequency (Hz)

500

1000

1500

2000

2500

3000

3500

4000

Amplitude

Figure 1.3: Spectrum of a segment from the violin recording.

The lowest frequency component is called the fundamental frequency. The

fundamental frequency of this signal is near 440 Hz (actually a little lower,

or “ﬂat”).

In this signal the fundamental frequency has the largest amplitude, so it is

also the dominant frequency. Normally the perceived pitch of a sound is

determined by the fundamental frequency, even if it is not dominant.

The other spikes in the spectrum are at frequencies 880, 1320, 1760, and

2200, which are integer multiples of the fundamental. These components

are called harmonics because they are musically harmonious with the fun-

damental:

• 880 is the frequency of A5, one octave higher than the fundamental.

An octave is a doubling in frequency.

• 1320 is approximately E6, which is a perfect ﬁfth above A5. If you

are not familiar with musical intervals like "perfect ﬁfth”, see

https:

//en.wikipedia.org/wiki/Interval_(music)

• 1760 is A6, two octaves above the fundamental.

• 2200 is approximately C]7, which is a major third above A6.

These harmonics make up the notes of an A major chord, although not all

in the same octave. Some of them are only approximate because the notes

that make up Western music have been adjusted for equal temperament

(see

http://en.wikipedia.org/wiki/Equal_temperament

1.3. Signals 5

Given the harmonics and their amplitudes, you can reconstruct the signal

by adding up sinusoids. Next we’ll see how.

1.3 Signals

I wrote a Python module called

thinkdsp.py

that contains classes and func-

tions for working with signals and spectrums

. You will ﬁnd it in the repos-

itory for this book (see Section 0.2).

To represent signals,

thinkdsp

provides a class called

Signal

, which is the

parent class for several signal types, including

Sinusoid

, which represents

both sine and cosine signals.

thinkdsp

provides functions to create sine and cosine signals:

cos_sig = thinkdsp.CosSignal(freq=440, amp=1.0, offset=0)

sin_sig = thinkdsp.SinSignal(freq=880, amp=0.5, offset=0)

freq

is frequency in Hz.

amp

is amplitude in unspeciﬁed units where 1.0 is

deﬁned as the largest amplitude we can record or play back.

offset

is a phase offset in radians. Phase offset determines where in the

period the signal starts. For example, a sine signal with

offset=0

starts at

sin 0, which is 0. With

offset=pi/2

it starts at sin π/2, which is 1.

Signals have an

__add__

method, so you can use the

operator to add them:

mix = sin_sig + cos_sig

The result is a

SumSignal

, which represents the sum of two or more signals.

A Signal is basically a Python representation of a mathematical function.

Most signals are deﬁned for all values of

, from negative inﬁnity to inﬁnity.

You can’t do much with a Signal until you evaluate it. In this context, “eval-

uate” means taking a sequence of points in time,

, and computing the

corresponding values of the signal,

. I represent

and

using NumPy

arrays and encapsulate them in an object called a Wave.

A Wave represents a signal evaluated at a sequence of points in time. Each

point in time is called a frame (a term borrowed from movies and video).

The measurement itself is called a sample, although “frame” and “sample”

are sometimes used interchangeably.

The plural of “spectrum” is often written “spectra”, but I prefer to use standard English

plurals. If you are familiar with “spectra”, I hope my choice doesn’t sound too strange.

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