离散时间信号处理

– Processing analog signals in a digital way. – Processing digital signals.
• Advantages:
– High reliability（可靠性） – High agility (灵活性好，易于实现系统性能） – High precision（高精度） – Low cost （成本低） –…
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Example: generate the signal with impulse sequence
a3
a3 (n 3)2 -1 0 1 2 3 4 5 n
a6
n 0
a2 (n 2)
a2
n 0
a6 (n 6)
0
x(n) a3 (n 3) a2 (n 2) a6 (n 6)
A convenient notation for the sequence x just is x(n).
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2.1 Discrete-time signals — graph
• Discrete-time signals are often depicted graphically.
• A discrete-time signal can be represented as
{x(nT), n Z}
where T is time interval between samples. Each sample of sequence x(nT) is determined by the amplitude of signal at instant nT. For example
– Discrete-time signals are those that are defined at discrete times.
Amplitude Amplitude
t Continue-time signal
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t
Discrete-time signal
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2.1 Discrete-time signals — notations
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Introduction
• What is Digital Signal Processing?
– Digital Signal Processing is the science to process signals by digital means.
– This includes a wide variety of goals: filtering, transformation, recognition, enhancement, compression, and much more.
x(n) or x(nT)
n or nT
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unit sample sequence（单位抽样序列）
• The definition of the unit impulse
(n)
1, 0,
n0 n0
δ(n) 1
0
n
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delayed unit sample sequence （延时单位抽样序列）
– DSP is one of the most powerful technologies that will shape science and engineering in the twenty-first century. Suppose we attach an analog-to-digital converter to a computer, and then use it to acquire a chunk of real world data.
{x(n), n Z} where Z is the set of integer numbers（整数集）, and x(n) is referred to as the “nth sample” of the sequence. For example
..., x(0), x(1), x(2), x(3),...
cos(2 n 16)
x(n)
0
n
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Exponential Sequence （实指数序列）
• The definition of the real exponential function is
x(n) A n
αn
0
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n
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unit ramp（单位斜坡序列）
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References
• 程佩青，数字信号处理教程，清华大学出版社
• 胡广书，数字信号处理——理论、算法与实现，
清华大学出版社
• 高西全，丁玉美，阔永红，数字信号处理——原
理、实现及应用，电子工业出版社
•…
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Chapter 2 Discrete-Time Signals and System
冲激响应）&FIR （有限长单位冲激响应）filter). – Multirate systems（多率系统）, filter banks（滤波器组）,
and wavelets（小波）. – Implementation of digital signal processing systems.
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a picture would be represented mathematically as a brightness function of two spatial variables------f(x,y). ------ Two-dimensional (2-D) signal 二维信号
a color video signal (a RGB television signal) is a 3-D signal. -----Multidimensional (M-D) signal 多维信号
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2.1 Discrete-Time Signals: Sequences
• The independent variable of a signal may be either continuous or discrete.
– Continuous-time signals are those that are defined at continuous times.
• The definition of the delayed unit sample sequence
(n
m)
1, 0,
nm nm
δ(n – m) 1
0
m
n
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unit step sequence （单位阶跃序列）
• The definition of the unit step
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Introduction
• 简单说，数字信号处理是利用计算机或专用处 理设备，以数值计算方法对信号进行采集 、变 换、综合、估值与识别等加工处理，借以到达 提取信息和便于应用目标。
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Introduction
• Digital signal processing includes two meanings:
u(n)
1, 0,
n0 n0
u(n) 1
0
n
——u(n)后向差分
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cosine function
• The definition of the cosine function is x(n) cos(n) ,
whose angular frequency（角频率） is ω rad/sample.
radar
Audio signal processing
Communication system
digital TV
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Multimedia system
Image processing
Mobile telephone
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Introduction
• Signals
– A signal can be defined as a function that conveys information.
– If 2π/ω is an irrational number（无理数）, this cosine function will not be periodic at all.
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Example
• Determine whether following discrete signal is periodic or not. If it is, calculate the period of the signal.
• Note, not all discrete cosine functions are periodic.
cos(n) cos((n N )), n Z ωN 2k
N 2 k
– If 2π/ω is an integer（整数） or a rational number（有理数), this sequence will be periodic;
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Introduction
Main tools: – Discrete-time signal representations. – Discrete transforms and their fast algorithms (z transform,
DFT&FFT). – Design and implementation of digital filter (IIR（无限长单位
• The definition of the unit ramp n, n 0
r(n) 0, n 0
r(n)
0
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n
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2.1 Discrete-time signals
• An arbitrary sequence can be expressed as a sum of scaled, delayed unit impulses.
x(n) x(k) (n k) k
• The unit step u(n) can be expressed as
u(n) (n k) k 0
• And the unit ramp r(n) can be expressed as
r(n) n (n k) k 0
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..., x(0), x(0.03), x(0.06), x(0.09),...
where T is 0.03. • Another notation is a sequence of numbers. For example, the
sequence x can be represented as
– Signals are presented mathematically as functions of one or more independent variables.
for example:
a speech signal would be represented mathematically as a function of one time variable------f(t); ----- One-dimensional (1-D) signal 一维信号
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n
a6
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Periodic sequence
• A sequence x(n) is defined to be periodic if and only if there is an integer N≠0 such that x(n) = x(n + N) for all n. In such a case, N is called the period of the sequence.
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Introduction
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Introduction
• Digital Signal Processing (DSP) is used in a wide variety of applications.
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