Note: This blog is based on the second edition of Oppenheim's "Signals and Systems", mainly for the review and deepening of their own learning.
1. Discrete-time processing of continuous-time signals
1. In many applications, a continuous time signal is first converted into a discrete time signal, then processed, and then converted into a continuous time signal after processing. Broadly speaking, the compromise processing method for continuous-time signals can be seen as a cascade of three links as shown in the figure below, where xc(t) and yc(t) are both continuous-time signals, and xd( Both t) and yd(t) are corresponding discrete time signals.
Through a process of periodic sampling (its sampling frequency satisfies the conditions in the sampling theorem), the continuous time series xc(t) can be completely represented by an instantaneous sample value xc(nT); that is, the discrete time series xd [n] The following formula
I want to connect with xc(t). Transforming xc(t) to xd[n] corresponds to the first system in the figure above, called the continuous-time-to-discrete-time transformation (C/D), while the third system is the inverse transformation to the above, i.e. Discrete-time to continuous-time conversion (D/C). What D/C implements is the interpolation between the sample points of its input; that is, after D/C, a continuous-time signal yc(t) is generated, which yc(t) and its input discrete-time signal yd(t) is related by the following formula
This concept is more clearly represented in the diagram below.
In order to study the relationship between the continuous-time signal xc(t) and its discrete-time representation xd(t), the transformation from continuous-time to discrete-time can be expressed as a process of periodic sampling, followed by an impulse A string is mapped into a sequence link. Both steps are represented in the following figure
The first step in the figure represents a sampling process, the impulse train xp(t) is an impulse sequence, the amplitude of each impulse train corresponds to the sample value of xc(t), and the time interval is equal to the sampling period T . Then, in the conversion from impulse train to discrete time series, xd(t) is obtained; this is the same sequence as the castrated version of xc(t), but with a new independent variable n for the unit interval. Therefore, in practice the conversion from impulse trains of samples to discrete time series of samples can be considered as a time normalization process.
2. It is also meaningful to examine the transformation from the perspective of the frequency domain. The frequency variable of continuous time is represented by w, and the frequency variable of discrete time is represented by w 
The continuous-time Borier transform Xp(t) of xp(t) is represented by the sample values of xc(t). because
According to 
Now consider the discrete-time Borier transform of xd[n], i.e.
Alternatively, there is
Comparing Equation (7.18) and Equation (7.20) , it can be seen that they
in addition
thus get
3. A continuous-time impulse string yp(t) can be generated from the sequence yn[n], and the recovery of the two continuous-time yc(t) can be realized by the low-pass filtering method shown in the figure below
Now, with the entire system shown below
It is clear that if the discrete-time system in the figure is an identity system, and Jiading satisfies the conditions in the sampling theorem, then the whole system must be an identity system.
1), digital differentiator
1. Now consider a discrete-time implementation of a continuous-time band-limited breeze. The frequency response of the continuous time differential filter is
The frequency response from a band-limited breeze with cutoff frequency wc is
As shown below
If ws=2wc, the corresponding discrete-time frequency response 
As shown below
Using the discrete-time frequency response, yc(t) must be the derivative of xc(t) as long as no aliasing occurs in the sampling of cx(t).
2), half sampling interval delay
1. In order to realize the time shift (delay) problem of a continuous time signal. Therefore, according to the requirements, the input xc(t) is band-limited, and the sampling rate is higher than one to avoid aliasing, the input and output of the entire system are connected by the following relationship;
According to the time-shifting property, there are
An equivalent continuous-time system that actually wants to love you must be band-limited, so choose
That is to say, Hc(jw) corresponds to a time shift of Eq. (7.33) for signals within the band limit, and all frequencies higher than wc are filtered out. The mode and phase characteristics of this frequency response are shown below.
If the sampling frequency ws=2wc is taken, the corresponding discrete-time frequency response 
For an appropriate band-limited input, if it 
If Δ/T is not an integer, equation (7.36) has no meaning, since the sequence is only defined over integer n values. However, we can use band-limited interpolation to explain the relationship between xd[n] and yd[n] in these cases. The wash numbers xc(t) and xd[n] are linked by sampling and band-limited interpolation, as are yc(t) and yd[n]. If 
This condition is sometimes referred to as a half-sample interval delay.
2. Discrete-time signal sampling
1), pulse train sampling
1. Similar to continuous time sampling, the sampling of discrete time signals can also be expressed as the system shown in the figure below
Here, the new sequence xp[n] formed by the sampling process is equal to the original sequence x[n] at an integer multiple of the sampling period N, and is zero between the sampling points, that is,
The frequency domain effect of discrete-time sampling can be derived using the multiplicative property. So, since
in the frequency domain
The Borier transform of the sampling sequence p[n] is
In the formula, the sampling frequency ws=2π/N. Combining formula (7.40) and formula (7.41), we get
2. Fully recover a discrete-time signal from the sample using an ideal low-pass filter
Similar to continuous-time sampling, the two sequences x[n] and xr[n] are always equal at integer multiples of the sampling period; that is,
This has nothing to do with aliasing.
The process of reconstructing x[n] by applying a low-pass filter to xp[n] can also be regarded as an interpolation formula in the time domain. Using h[n] to represent the unit impulse response of the low-pass filter, we have
The reconstructed sequence xr[n] is
or equivalently written as
The above equation represents an ideal band-limited interpolation, thus requiring the realization of an ideal low-pass filter. In general applications, a properly approximated low-pass filter is often used. At this time, the equivalent interpolation formula has the following formula
where hr[n] is the unit impulse response of the interpolation filter. Similar to continuous-time interpolation, in discrete-time interpolation, there are interpolation approximations such as zero-order hold and first-order hold.
2), discrete-time extraction and interpolation
1. The principle of discrete-time sampling has many, many important applications such as filter design and implementation of fire communications. For discrete time sampling, the sequence is often replaced by a new sequence xb[n], and xb[n] is composed of the sequence values at every N point in xp[n], that is
Alternatively, since both xp[n] and x[n] are equal over integer multiples of N, it can be equivalent to
Generally, the process of extracting samples at every Nth point is called extraction. The relationship between x[n], xp[n] and xb[n] is shown in the following figure
2. In order to determine the effect of extraction in the frequency domain, it is hoped that the relationship between the Borier transforms of xb[n] can be obtained 
Using equation (7.48), we have
If n=kN, or k=n/N, then it can be written as
Because when n is not an integer multiple of N, xp[n]=0, so the above formula can also be written as
Furthermore, the right side of equation (7.52) is the Borier transform of xp[n], namely
Therefore, from equations (7.52) and (7.53), we can get
This relationship is shown in the figure below
2. If the original sequence x[n] is obtained by sampling the continuous time signal of essential oils, the extraction process can be regarded as the result of reducing the sampling rate to the original 1/N on the continuous time signal. Therefore, in order to avoid aliasing during the decimation process, X(ejw) of the metasequence x[n] cannot cover the entire frequency band. In other words, if the sequence can be de-silked without introducing aliasing, then the original continuous-time signal is ten times sampled, so that the meta-sampling rate can be reduced without introducing aliasing. Therefore, the process of decimation is often referred to as intersampling.
Just as intersampling is useful in writing applications, there are situations in which it is necessary to convert a sequence to a higher equivalent sampling rate, a process called upsampling or interpolation. Upsampling is basically the inverse process of decimation or downsampling.