1. Chapter 2 Discrete-Time Signals and Systems
Content
The Discrete-Time Signal: Sequence
The Discrete-Time System
The Discrete-Time Fourier Transform (DTFT)
The Symmetric Properties of the DTFT
System Function and Frequency Response
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2. The Discrete-Time Signal: Sequences
Elementary sequences
Unit sample sequence
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3. The Discrete-Time Signal: Sequences
Unit step sequence
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4. The Discrete-Time Signal: Sequences
Rectangular sequence
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5. The Discrete-Time Signal: Sequences
Sinusoidal sequence
amplitude
digital angular frequency
phase
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6. The Discrete-Time Signal: Sequences
Real-valued exponential sequence
The is convergent when
The is divergent when
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7. The Discrete-Time Signal: Sequences
Complex-valued exponential sequence
Attenuation factor=0,>0,<0时，幅度保持不变，变大，变小。
Attenuation factor
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8. The Discrete-Time Signal: Sequences
Classification of sequences
Finite-length sequence
is defined only for a finite time interval:
where
examples
至少是除z=0和z=无穷的开域（0，无穷）“有限z平面”。
当N1≥0，0<|z|≤无穷，
当N2≤0，0≤|z|<无穷，
The length of a finite-length sequence can be increased by zero-padding
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9. The Discrete-Time Signal: Sequences
Right-sided sequence
has zero-valued samples for
where
If , a right-sided sequence is called a causal sequence
右边序列z变换的收敛域为：Rx_<|z|<“无穷”，Rx_是收敛域的最小半径。对于因果序列，在“无穷”处也收敛。
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10. The Discrete-Time Signal: Sequences
Left-sided sequence
has zero-valued samples for
where
If , a left-sided sequence is called a anti-causal sequence
左边序列z变换的收敛域为：0<|z|<Rx+，Rx+是收敛域的最大半径。对于反因果序列，在0处也收敛。
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11. The Discrete-Time Signal: Sequences
Two-sided sequence
is defined for any n
a dual-sided sequence can be seen as the sum of a right-sided sequence and a left-sided sequence.
双边序列的收敛域：Rx_<|z|<Rx+
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12. The Discrete-Time Signal: Sequences
Absolutely summable sequence
Example:
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13. The Discrete-Time Signal: Sequences
Square-summable sequence
Example:
It is square-summable but not absolutely summable
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14. The Discrete-Time Signal: Sequences
Operations on sequence
Time-shifting operation
where is an integer
delaying operation
z-1
Unit delay
advance operation z
Unit advance
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15. The Discrete-Time Signal: Sequences
Time-reversal (folding) operation
Addition operation
Sample-by-sample addition
时间翻褶
Adder
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16. The Discrete-Time Signal: Sequences
Scaling operation
Multiplier A
Product (modulation) operation
Sample-by-sample multiplication
Scaling：缩放比例
modulator
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17. The Discrete-Time Signal: Sequences
Sample summation
Sample production
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18. The Discrete-Time Signal: Sequences
Sequence energy
Sequence power
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19. The Discrete-Time Signal: Sequences
Decimation by a factor D
Every D-th samples of the input sequence are kept and others are removed:
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20. The Discrete-Time Signal: Sequences
Interpolation by a factor I
I -1 equidistant zeros-valued samples are inserted between each two consecutive samples of the input sequence.
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21. The Discrete-Time Signal: Sequences
The periodicity of sequence if
: any integer
: positive integer
then the is called a periodic sequence,
and the value of N is called the fundamental period.
提问：正弦序列是否一定是周期性的，其周期是多少？
a periodic sequence is usually expressed as
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22. The Discrete-Time Signal: Sequences
The periodicity of sinusoidal sequence If
, : any integer
is a periodic sequence and its period is
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23. Copyright © 2005. Shi Ping CUC
If is a integer
If is a noninteger rational number
If is a irrational number
is an aperiodic sequence
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24. The Discrete-Time Signal: Sequences
The periodicity of Complex-valued exponential sequence
when , the periodicity of Complex-valued exponential sequence is the same as the sinusoidal sequence
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25. The Discrete-Time Signal: Sequences
The periodicity of sinusoidal sequence which is developed by uniformly sampling a continuous-time sinusoidal signal
Analog angular frequency
连续时间的正弦信号本身具有周期性，对其抽样后所得到的正弦序列的周期与原来的正弦信号的周期有何关系？
Sampling period
Sampling angular frequency
Digital angular frequency
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26. The Discrete-Time Signal: Sequences
Units:
Sampling period : seconds/sample
Analog frequency : hertz (Hz)
Analog angular frequency : radians/second
Digital angular frequency : radians/sample
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27. The Discrete-Time Signal: Sequences
The periodicity:
The period of the continuous-time sinusoidal signal
The sampling period
If is a rational number, then
are positive integers
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28. The Discrete-Time Signal: Sequences
Sequence synthesis
Unit sample synthesis
Any arbitrary sequence can be synthesized in the time-domain as a weighted sum of delayed (advanced) and scaled unit sample sequence.
单位取样序列的移位加权和
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29. The Discrete-Time Signal: Sequences
Even and odd synthesis
Even (symmetric):
Odd (antisymmetric):
Any arbitrary real-valued sequence can be decomposed into its even and odd component:
return
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30. The Discrete-Time System
Introduction
A discrete-time system processes a given input sequence x(n) to generate an output sequence y(n) with more desirable properties.
Mathematically, an operation T [ • ] is used.
y(n) = T [ x(n) ]
x(n): excitation, input signal
y(n): response, output signal
example
example
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31. The Discrete-Time System
Classification
Linear System
Time-Invariant (Shift-Invariant) System
Linear Time-Invariant (LTI) System
Causal System
Stable System
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32. The Discrete-Time System
Linear System
A system is called linear if it has two mathematical properties: homogeneity and additivity.
Accumulator
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33. The Discrete-Time System
Time-Invariant (Shift-Invariant) System
Accumulator
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34. The Discrete-Time System
Linear Time-Invariant (LTI) System
A system satisfying both the linearity and the time-invariance properties is called an LTI system.
LTI systems are mathematically easy to analyze and characterize, and consequently, easy to design.
A accumulator is an LTI system !
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35. The Discrete-Time System
The output of an LTI system is called
linear convolution sum
An LTI system is completely characterized in the time domain by the impulse response h(n).
example
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36. The Discrete-Time System
Causal System
In a causal system, the -th output sample depends only on input samples for and does not depend on input samples for
e.g.
如y(n)=x(n+1)即为非因果系统
For a causal system, changes in output samples do not precede changes in the input samples.
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37. The Discrete-Time System
An LTI system will be a causal system if and only if :
An ideal low-pass filter is not a causal system !
A sequence is called a causal sequence if :
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38. The Discrete-Time System
Stable System
A system is said to be bounded-input bounded-output (BIBO) stable if every bounded input produces a bounded output, i.e.
An LTI system will be a stable system if and only if :
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39. The Discrete-Time System
The M-point moving average filter is BIBO stable :
prove
A causal LTI discrete-time system:
证明参见Ch2（3）P5
以及Ch2（2）P28
prove
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40. The Discrete-Time System
Causal and Stable System
A system is said to be a causal and stable system if the impulse response is causal and absolutely summable , i.e.
return
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41. The Discrete-time Fourier Transform (DTFT)
The transform-domain representation of discrete-time signal
Discrete-Time Fourier Transorm (DTFT)
Discrete-Fourier Transform (DFT)
z-Transform
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42. The Discrete-time Fourier Transform (DTFT)
The definition of DTFT
DTFT:
IDTFT:
在黑板上推导反变换公式
在黑板上推导X(ejw)的收敛问题
Existence condition:
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43. The Discrete-time Fourier Transform (DTFT)
The comparison of vs.
Time domain
Frequency domain
discrete
continuous
Real valued
Complex-valued
Summation
integral
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44. The Discrete-time Fourier Transform (DTFT)
About
It is a periodic function of with a period of
The range of
The integral range of
It can be expressed as
X(ejw)也可以表示成实部和虚部。
magnitude function
phase function
and are all real function of
example
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45. The Discrete-time Fourier Transform (DTFT)
DTFT vs. z Transform
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46. The Discrete-time Fourier Transform (DTFT)
The general properties of DTFT
Linearity
The DTFT is a linear transformation
Time shifting
A shift in the time domain corresponds to the phase shifting
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47. The Discrete-time Fourier Transform (DTFT)
Frequency shifting
Multiplication by a complex exponential corresponds to a shift in the frequency domain
Convolution
Convolution in time domain corresponds to multiplication in frequency domain
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48. The Discrete-time Fourier Transform (DTFT)
Multiplication
Energy (Parseval’s Theorem)
energy density spectrum
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49. The Discrete-time Fourier Transform (DTFT)
Multiplied by an exponential sequence
Sequence weighting
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50. The Discrete-time Fourier Transform (DTFT)
Conjugation
Conjugation in the time domain corresponds to the folding and conjugation in the frequency domain
Folding
Folding in the time domain corresponds to the folding in the frequency domain
Conjugation and Folding
Conjugation and folding in the time domain corresponds to the conjugation in the frequency domain
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51. The symmetric properties of the DTFT
Conjugate symmetry of
Conjugate symmetric sequence:
For real-valued sequence, it is even symmetric:
Conjugate antisymmetric sequence:
证明：如果是共轭对称序列，则其实部偶对称，虚部奇对称
For real-valued sequence, it is odd symmetric:
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52. The symmetric properties of the DTFT
Any arbitrary sequence can be expressed as the sum of a conjugate symmetric sequence and a conjugate antisymmetric sequence
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53. The symmetric properties of the DTFT
Conjugate symmetry of
The can be expressed as the sum of the conjugate symmetric component and the conjugate antisymmetric component
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54. The symmetric properties of the DTFT
conjugate symmetric
For real-valued function, it is even symmetric
conjugate antisymmetric
For real-valued function, it is odd symmetric
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55. The symmetric properties of the DTFT
Implication: If the sequence is real and even, then is also real and even.
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56. The symmetric properties of the DTFT
If the sequence x(n) is real, then
example
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57. The symmetric properties of the DTFT
The DTFT of periodic sequences
The DTFT of complex-valued exponential sequences
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58. The symmetric properties of the DTFT
The DTFT of constant-value sequences
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59. The symmetric properties of the DTFT
The DTFT of unit sample sequences
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60. The symmetric properties of the DTFT
The DTFT of general periodic sequences
return
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61. System Function and Frequency Response
The representation of a LTI system
Impulse response
Difference equation
System function
这里指的是离散时间LTI系统
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62. System Function and Frequency Response
System function (Transfer function)
The z-transform of the impulse response h(n) of the LTI system is called system function or transfer function
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63. System Function and Frequency Response
The region of convergence (ROC) for H(z)
An LTI system is stable if and only if the unit circle is in the ROC of H(z)
An LTI system is causal if and only if the ROC of H(z) is
An LTI system is both stable and causal if and only if the H(z) has all its poles inside the unit circle, i.e. the ROC of H(z) is
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64. System Function and Frequency Response
System function vs. difference equation
difference equation
take z-transform for both sides
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65. System Function and Frequency Response
Frequency response of an LTI system
The DTFT of an impulse response is called the frequency response of an LTI system, i.e.
注意，h(n)为实数序列。因此H(exp(jw))为共轭偶对程，即幅度偶对称，相位奇对称。
example
magnitude response function
example
phase response function
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66. System Function and Frequency Response
Group delays
In general, the frequency response is a complex function of
介绍群延时的含义
is a continuous function of
is a periodic function of , the period is
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67. System Function and Frequency Response
Response to exponential sequence
The output sequence is the input exponential sequence modified by the response of the system at frequency
ejw0n
称为系统的特征函数
H(ejw0)称为特征值（即：系统在w0处的频率响应）
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68. System Function and Frequency Response
Response to sinusoidal sequences
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69. System Function and Frequency Response
Response to arbitrary sequences
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70. System Function and Frequency Response
Geometric interpretation of frequency response
这一部分内容可演示pez from dspfirst.
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71. System Function and Frequency Response
is a real number
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72. System Function and Frequency Response
K有可能是负数，因此需要表示成模和相角的形式。
zero vector
pole vector
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73. System Function and Frequency Response
zero vector
pole vector
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74. System Function and Frequency Response
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75. System Function and Frequency Response
An approximate plot of the magnitude and phase responses of the system function of an LTI system can be developed by examining the pole and zero locations
To highly attenuate signal components in a specified frequency range, we need to place zeros very close to or on the unit circle in this range
To highly emphasize signal components in a specified frequency range, we need to place poles very close to or on the unit circle in this range
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76. System Function and Frequency Response
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77. System Function and Frequency Response
Minimum-Phase and Maximum-Phase system
the number of zeros inside the unit circle
the number of zeros outside the unit circle
the number of poles inside the unit circle
the number of poles outside the unit circle
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78. Copyright © 2005. Shi Ping CUC
A causal stable system
A causal stable system with all zeros inside the unit circle is called a minimum-phase delayed system
当omiga变化2pi时，单位圆内的零矢量或极矢量的相位也变化2pi；而单位圆外的零矢量或极矢量的相位变化量为0。
A causal stable system with all zeros outside the unit circle is called a maximum-phase delayed system
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79. Copyright © 2005. Shi Ping CUC
An anti-causal stable system
An anti-causal stable system with all zeros inside the unit circle is called a maximum-phase advanced system
An anti-causal stable system with all zeros outside the unit circle is called a minimum-phase advanced system
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80. System Function and Frequency Response
Important properties of minimum-phase delayed system
minimum-phase delayed system is often called minimum-phase system for short.
It plays an important role in telecommunications
Any nonminimum-phase system can be expressed as the product of a minimum-phase system function and a stable all-pass system
For all systems with the identical
当h(n)和hmin(n)都是N点序列时，当其总能量相同时，最小相位系统的能量更集中在n=0附近。
根据Pasvel定理，当两个序列的|H(ejw)|相同时，说明其频域能量相同，因而其时域能量也相同。所以上面的第一个式子是成立的。
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81. System Function and Frequency Response
All-pass system
Definition
A system that has a constant magnitude response for all frequencies, that is,
The simplest example of an all-pass system is a pure delay system with system function
This system passes all signals without modification except for a delay of k samples.
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82. System Function and Frequency Response
1-th order all-pass system
在黑板上画全通系统的零极点镜像对称关系
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83. System Function and Frequency Response
An alternative form of 1-th order all-pass system
先取共轭再求倒数＝先求倒数再取共轭
Mirror image symmetry with respect to the unit circle
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84. System Function and Frequency Response
2-th order all-pass system
选择两个零点（两个极点）互为共轭是为了保证多项式系数为实数。
example
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85. System Function and Frequency Response
example
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86. System Function and Frequency Response
N-th order all-pass system
因为D(z)的系数为实数，可看成是实序列的z变换，因此其频响共轭翻转。教材P73 表中No.16
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87. System Function and Frequency Response
An alternative form for N-th order all-pass system
The number of real poles and zeros
The number of complex-conjugate pair of poles and zeros
For causal and stable system
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88. System Function and Frequency Response
Application
Phase equalizers
When placed in cascade with a system that has an undesired phase response, a phase equalizer is designed to compensate for the poor phase characteristics of the system and therefore to produce an overall linear-phase response.
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89. System Function and Frequency Response
Group delays
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90. System Function and Frequency Response
Any causal-stable nonminimum-phase system can be expressed as the product of a minimum-phase delayed system cascaded with a stable all-pass system
example
a minimum-phase system
a pair of conjugate zeros outside the unit circle
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91. Copyright © 2005. Shi Ping CUC
is a minimum-phase system
is a 2-th all-pass system
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92. System Function and Frequency Response
By cascading an all-pass system an unstable system can be made stable without changing its magnitude response
example
这里指的是因果稳定系统
unstable system
stable system
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93. System Function and Frequency Response
Relationships between system representations
Difference
Equation
Express H(z) in z-1 cross multiply and take inverse
Inverse ZT ZT
take ZT solve for Y/X
虚线表示系统要稳定.
substitute
Inverse DTFT
DTFT
Take DTFT solve for Y/X
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Rectangular sequence
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98. Real-valued exponential sequence
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99. Complex-valued exponential sequence
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106. Time-shifting operation
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folding operation
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addition operation
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modulation operation
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periodic sequence
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Periodicity of sequence
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112. Periodicity of sequence 2
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115. Copyright © 2005. Shi Ping CUC
Accumulator
The output at time instant n is the sum of the input sample at time instant n and the previous output at time instant n-1, which is the sum of all previous input sample values from to n-1
The input-output relation can also be written in the form:
This form is used for a causal input sequence, in which case y(-1) is called the initial condition
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M-point moving-average system
An application: consider
Where is the signal, and is a random noise
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Accumulator
Hence, the above system is linear
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Accumulator
Hence, the above system is time-invariant
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The M-point moving average filter
For a bounded input , we have
Hence, the M-point moving average filter is BIBO stable
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A causal LTI discrete-time system
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Mirror image symmetry
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