Table of contents
1.3 Sampling of continuous-time signals
2. Spectrum of an ideal sampled signal
4. Reconstruct the band-limited signal from the sampled signal sequence
1.4 Time Domain Analysis of Discrete Time Systems
Discrete Time System Representation
2. Time-invariant systems (shift-invariant systems)
3. The relationship between the unit sampling response and the input and output of the system
4. Properties of linear time-invariant systems
1.5 Linear Difference Equations with Constant Coefficients
1.3 Sampling of continuous-time signals
Sampling of a continuous-time signal is the process of converting a continuous-time signal to a discrete-time signal:
1. Ideal sampling
In ideal sampling, the pulse function should have zero width, infinite amplitude, and zero spacing between sample sequences. This makes the sampled discrete-time signal immune to the sampling process and enables a complete reconstruction of the original continuous-time signal.
However, ideal sampling is a theoretical concept that cannot be fully realized in practice. In practice, there are factors such as the limited bandwidth of the sampler and the actual response of the anti-aliasing filter in the sampling process, resulting in a certain difference between the sampled signal and the original signal. Therefore, in practical applications, it is more common practice to use an approximate sampling method combined with a suitable anti-aliasing filter for signal reconstruction.
2. Spectrum of an ideal sampled signal
Linear time-invariant system: multiplication in the time domain, convolution operation in the frequency domain! ! !
What happens to the spectrum after ideal sampling?
Folding frequency: half of the sampling frequency
Nyquist sampling theorem! ! !
In order to restore the original signal without distortion after sampling, the sampling frequency must be greater than twice the highest frequency of the signal spectrum.
(If the highest frequency of a signal is f_max, the sampling rate fs must satisfy fs ≥ 2*f_max)
The significance of the sampling rate is to determine the number of times the signal is sampled within a specific period of time. If the sampling rate is too low, the information between sampling points will be lost, that is, aliasing will occur, and the original signal cannot be accurately reconstructed.
3. Recovery of samples
Sampling recovery refers to reconstructing the original continuous-time signal as accurately as possible by processing the sampled discrete-time signal.
In ideal sampling that satisfies Nyquist's theorem, the sampled spectrum does not produce spectral aliasing:
4. Reconstruct the band-limited signal from the sampled signal sequence
interpolation function
1.4 Time Domain Analysis of Discrete Time Systems
Discrete Time System Representation
The most commonly used and important of discrete-time systems is the " linear time-invariant system "
1. Linear system
Linear System: A system that satisfies the principle of superposition is called a linear system.
The principle of superposition refers to the weighted linear combination of the input signal of the system, and its output signal is also the relationship of the corresponding weighted linear combination.
- Linear systems must satisfy both superposition and homogeneity ;
- Signal and proportionality constants can be complex numbers;
- A linear system with zero input produces zero output;
- The linear system is the basis of "decomposition of signals and superposition of responses"
example
Note: A system represented by a linear equation is not necessarily a linear system .
2. Time-invariant systems (shift-invariant systems)
Refers to the fact that the behavior of the system does not change over time. In other words, for a given input signal, the response of the system is the same at different points in time.
Examples
3. The relationship between the unit sampling response and the input and output of the system
4. Properties of linear time-invariant systems
5. Causal system
Causal system: the output of the system at any moment depends only on the input at that moment and before that moment ( the response of the system is determined by past and current input signals, and will not be affected by future input signals )
Non-causal system: the output of the system at any time depends not only on the input at that time and before that time, but also on the input in the future
6. Stabilize the system
Stable system: Bounded input, produces bounded output. ( BIBO )
A stable system means that within a limited input range, the output of the system will remain bounded or tend to a bounded area. In other words, stable systems do not exhibit infinitely growing or diverging outputs.
1.5 Linear Difference Equations with Constant Coefficients
Constant Coefficient Linear Differential Equation - Describes the input-output relationship of a continuous-time linear time-invariant system
Constant Coefficient Linear Difference Equation — Describing the Input-Output Relationship of a Discrete-Time Linear Time-Invariant System
Difference Equation Representation of Discrete Systems
- Easy to get the system structure directly
- It is convenient to solve the transient response of the system
Constant Coefficient Linear Difference Equation Solving
1. Time domain method
- Iterative method (simple, not easy to get a closed solution)
- Convolution method (used to solve the zero-state response of the system)
2. Transform domain method z-transform
