Some key points
1,Any arbitrary input sequence x[n] can be expressed as a linear combination of delayed and advanced unit sample sequences
2,?Linear Time-Invariant (LTI) System
A system satisfying both the linearity and the time-invariance property.
If y1[n] is the output due to an input x1[n] and y2[n] is the output due to an input x2[n] then for an input
x[n]=ax1[n]+bx2[n]
the output is given by
y[n]=ay1[n]+by2[n]
Hence,the above system is linear,
Above property must hold for any arbitrary constants a and b and for all possible inputs x1[n] and x2[n].
For a shift-invariant system,if y1[n] is the response to an input x1[n],then the response to an input
x[n]=x1[n-n0]
is simply
y[n]=y1[n-n0]
where n0 is any positive or negative integer
The above relation must hold for any arbitrary input and its corresponding output
The above property is called time-invariance property,or shift-invariant proterty
-
For example,Down-sampling is time-variant
3．Convolution in time domain
If a signal inputs to a system with the impulse response,the output signal
The main idea of the method is that an input signal is represented as(decomposed) the sum of basic signal:,the response of LTI system is the synthesis of basic response,
4,Discrete-Time Signals In the Transform Domain
(1) From FT TO DTFT and DFT
The DTFT X(ej() of a sequence x[n] is a continuous function ofIt is also a periodic function of? with a period 2DTFT is the Fourier Transform of discrete-time sequence,It is discrete in time domain and its spectrum is periodical.
(2) Relationships ZT and DTFT and DFT
A finite-length sequence
.
when ,
That means,The ZT on the unit circle in Z-plane is the DTFT of ,The samples on the unit circle in Z-plane,,are the DFT of .
5,The Concept of Filtering
One application of an LTI discrete-time system is to pass certain frequency components in an input sequence without any distortion (if possible) and to block other frequency components。In another words,the sinusoidal components of the input,some of these components can be selectively heavily attenuated or filtered with respect to the others。
For example,a signal is inputted to a lowpass filter.If we change the frequency response of the filter,the output signal will be changed(as shown in following figure),
6,Analog Lowpass Filter Specifications
passband edge frequency：
stopband edge frequency：
Peak passband ripple,
Minimum stopband attenuation：
1,Any arbitrary input sequence x[n] can be expressed as a linear combination of delayed and advanced unit sample sequences
2,?Linear Time-Invariant (LTI) System
A system satisfying both the linearity and the time-invariance property.
If y1[n] is the output due to an input x1[n] and y2[n] is the output due to an input x2[n] then for an input
x[n]=ax1[n]+bx2[n]
the output is given by
y[n]=ay1[n]+by2[n]
Hence,the above system is linear,
Above property must hold for any arbitrary constants a and b and for all possible inputs x1[n] and x2[n].
For a shift-invariant system,if y1[n] is the response to an input x1[n],then the response to an input
x[n]=x1[n-n0]
is simply
y[n]=y1[n-n0]
where n0 is any positive or negative integer
The above relation must hold for any arbitrary input and its corresponding output
The above property is called time-invariance property,or shift-invariant proterty
-
For example,Down-sampling is time-variant
3．Convolution in time domain
If a signal inputs to a system with the impulse response,the output signal
The main idea of the method is that an input signal is represented as(decomposed) the sum of basic signal:,the response of LTI system is the synthesis of basic response,
4,Discrete-Time Signals In the Transform Domain
(1) From FT TO DTFT and DFT
The DTFT X(ej() of a sequence x[n] is a continuous function ofIt is also a periodic function of? with a period 2DTFT is the Fourier Transform of discrete-time sequence,It is discrete in time domain and its spectrum is periodical.
(2) Relationships ZT and DTFT and DFT
A finite-length sequence
.
when ,
That means,The ZT on the unit circle in Z-plane is the DTFT of ,The samples on the unit circle in Z-plane,,are the DFT of .
5,The Concept of Filtering
One application of an LTI discrete-time system is to pass certain frequency components in an input sequence without any distortion (if possible) and to block other frequency components。In another words,the sinusoidal components of the input,some of these components can be selectively heavily attenuated or filtered with respect to the others。
For example,a signal is inputted to a lowpass filter.If we change the frequency response of the filter,the output signal will be changed(as shown in following figure),
6,Analog Lowpass Filter Specifications
passband edge frequency：
stopband edge frequency：
Peak passband ripple,
Minimum stopband attenuation：
