This time I will not write out all the concepts in detail, but only focus on some key questions that are often asked in interviews.
Chapter 1 Basic Concepts of Signals and Systems
Here is an outline of the book Signals and Systems: This book studies the relationship between signals and systems.
1. What is the signal?
①Information is a form of expression in the natural world, such as sounds, breaths, images, etc. Signals are carriers of information, carry information, and change with the independent variable of time.
1. Classification of signals
(1) Continuous signals and discrete signals
If the value of the independent variable is continuous, it is a continuous signal; if both time and amplitude are continuous, it is called an analog signal.
If the independent variable has discontinuous values, it is called a discrete signal. A signal that is discrete in time but continuous in amplitude is called a sampling signal, and a signal that is discrete in time and discrete in amplitude is called a digital signal.
(2) Energy signal and power signal
① Energy signal: In an infinite time, the energy is limited, so the power is 0
② Power signal: In an infinite time, the power in each period is limited, so the energy tends to infinity.
2. Signal transformation
Mainly include signal addition, multiplication, differentiation, integration, deconvolution, time shift, scale transformation (scale reduction)
2. What is the system?
Broadly speaking: a system is a whole set of things with certain functions that are interconnected and restrictive.
Mathematically, a system is: something that acts on an input signal (excitation) to produce an output signal (response)
1. Classification of systems
(1) Linear systems and nonlinear systems
A linear system is a system that satisfies superposition and homogeneity.
Additivity means that the output obtained after multiple inputs act on the system is equivalent to the sum of the outputs of a single input acting on the system.
Homogeneity means that if the input becomes n times its original size, the output will also become n times its original size.
(2) Time-invariant and time-varying systems
are time-invariant: ① Component parameters do not change with time.
②The original input signal is input after a delay period, and the output signal after acting on the system will also have a corresponding delay.
(3) Causal and non-causal systems
Causal systems: Only causes have effects. The response of the system at a certain moment only depends on the input before this moment and has nothing to do with the input after this moment.
(4) Stable and unstable systems
Stable systems: Systems that only produce bounded outputs for any bounded input.
2. Core research object: LTI linear time-invariant system
3. What are the methods to analyze a system?
Answer: Mainly the input-output method and the state variable method.
Chapter 2 Time Domain Analysis of Continuous Time Signals and Systems - Differential Equations, Convolution
1. Several commonly used continuous time signals
① Real exponential signal
② Complex exponential signal
③ Sampling signal Sa (t) function: sint /t
④ Unit step function and unit impulse function
2. Continuous time system
Mainly remember these knowledge points:
①Continuous time functions can be represented by many unit impulse functions with different delays.
② For a system, a differential equation describing the characteristics of the system can be established. The mathematical model of the LTI system is a linear constant coefficient differential equation.
③ Solve a differential equation to obtain the response of the system. The total solution of the differential equation = homogeneous solution + special solution
④ Homogeneous solution - depends on the characteristics of the system itself - natural response
Special solution - related to the excitation signal - forced response
3.The importance of unit impulse response in LTI systems
① Since continuous time functions can be represented by many unit impulse functions with different delays, the unit impulse response can be used to conveniently solve the zero-state response of the system under any excitation signal.
②Unit impulse response is of great help in analyzing the causality and stability of the system.
4. What is the significance of convolution for signal and system analysis?
We know that any continuous time signal can be decomposed into the superposition of multiple impulse signals with different delays. Then we can know that the
zero-state response of a linear time-invariant system is the convolution of the input signal and the unit impulse response of the system .
Chapter 3 Frequency Domain Analysis of Continuous Time Signals and Systems – Fourier Transform
1. Fourier transform spectrum characteristics of general periodic signals
① Discreteness: The independent variables of the spectrum are discontinuous and discrete.
②Harmonicity: There are many harmonics in the spectrum, and harmonics only appear at integer multiples of the fundamental frequency Ω.
③Convergence: As the harmonic order increases, the amplitude of the harmonics will gradually decrease.
2. Frequency domain analysis of continuous time systems
1. Use Fourier transform to find the zero state response
This leads to the first method of analyzing signals and systems: the transformation domain analysis method, which converts the solution of differential equations (responses) in the time domain into the method of solving the zero-state response in the frequency domain.
Remember: the zero-state response r(t) of the system = the convolution of the excitation signal e(t) and the impulse response h(t) of the system r(t)
= e(t) * h(t)
① By the time domain convolution Product theorem: R(jw)=E(jw)H(jw)
②H(jw) is the system function, also called the frequency response function, which includes amplitude-frequency response and phase-frequency response.
③ Therefore, if a zero-state response is required, first find the Fourier transform of the excitation signal, then find the system function, find the convolution sum of the two, and then inversely transform to find the time domain response.
2. Conditions for system distortion-free transmission
①The amplitude-frequency characteristic of the system function H (jw) is a constant, that is, a straight line parallel to the x-axis.
②The phase-frequency characteristic of the system function H (jw) is a straight line passing through the origin.
Chapter 4 Complex Frequency Domain Analysis of Continuous Time Signals and Systems - Laplace Transform
1. Convergence region of Laplace transform
2. System functions
Similar to the Fourier transform, the Laplace transform of the system impulse response h(t) is the system function.
3. System stability discussion
①Stable: If all the poles of H(s) are in the left half plane, it is stable.
②Critically stable: If there is a single-level point at the origin or on the imaginary axis in addition to the left half plane, it is critically stable.
③As long as there is any pole in the right half plane, it will be unstable.
Chapter 5 Time Domain Analysis of Discrete Time Signals and Systems - Sampling Theorem
1. The concept of sampling
As we learned earlier, in order to convert an analog signal or a continuous signal into a discrete digital signal, we need: sampling - quantization - encoding.
Sampling: can change a signal with continuous amplitude in time domain into a signal with discrete amplitude and continuous time domain.
Quantization: can change a signal with continuous discrete amplitude in time domain into a signal with discrete time domain and amplitude.
2. Sampling theorem (Nyquist theorem)
For a signal, the highest frequency is Fm, and its spectrum diagram is some discrete impulse signals. There will be three situations in sampling:
If the sampling frequency F<2Fm, spectrum aliasing will occur.
If the sampling frequency F>=2Fm, spectrum aliasing will not occur and ideal sampling can be performed.
Call 2Fm the Nyquist sampling frequency
3. Discrete signals study sequences. It is different from the continuous time signal. The mathematical model of the continuous time signal and the system is a differential equation, while the mathematical model of the discrete time signal and the system is a difference equation.
There are also homogeneous solutions and special solutions, and there are also zero-state responses and zero-input responses.
4. Discrete-time systems also have convolutions, called convolution sums.
Similarly, the zero-state response of the system is the convolution sum of the excitation signal and the system impulse response.
Chapter 6 Z-domain analysis of discrete-time signals and systems—Z transform
1. Convergence region of Z transformation
Generally speaking, the convergence region of Z transformation can be divided into three situations:
Unilateral convergence: If the Z-transform sequence is a causal sequence, that is, starting from n=0 in the time domain, then the Z-transform converges within the unit circle (|z|<1). In the case of unilateral convergence, the Z transformation is resolved within the unit circle, and the convergence domain includes the unit circle and its interior.
Bilateral convergence: If the sequence of the Z transform is a bilateral sequence, that is, the sequence can have values before and after n=0 in the time domain, then the Z transform may converge in the entire complex plane. In the case of bilateral convergence, the Z transform is resolved in the entire complex plane, and the convergence domain is the entire complex plane.
Region convergence: In some special cases, the convergence region of the Z transformation may be a region on the complex plane. In this case, the shape and location of the convergence region depends on the specific sequence and the nature of the transformation. Generally speaking, the convergence region can be a sector, ring, rectangle or other shaped region on the complex plane.
2. Key points: The relationship between Fourier transform, Laplace transform and Z transform
Chapter 7 System State Variable Analysis Method
1. Methods related to the study of signals and systems: ① Input-output method ② State variable method
①Input-output method: For the analysis of the time domain and frequency domain of the system, whether it is solving differential equations, Fourier transform, Laplace transform, or Z transform, it is always aimed at an excitation signal and solving it in the system. response, which is a single input-single output form.
②State variable method: It is aimed at multi-input-multiple output systems and describes the state variables of the internal characteristics of the system.
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Summarize the possible interview questions for "Signals and Systems":
1. What is the subject of Signals and Systems studied?
Answer: As the name of the course "Signals and Systems" suggests, the main subject of the study is signals and systems, specifically deterministic signals (with known waveforms) and LTI linear time-invariant systems. A signal acts on a system and gets a response. The book uses two research methods: input-output method and state variable method to analyze system response.
2. What is the input and output method?
Answer: The input-output method is aimed at the output response of the input excitation signal under the action of the system. Most of this book talks about the input-output method, including time domain analysis and transform domain analysis. In the time domain, the initial state and differential equations of the system can generally be listed for solution; the transformation domain includes the frequency domain, complex frequency domain, and Z domain, which are embodied in Fourier transform, Laplace transform, and Z transform. By solving the system function to study the response.
3. What is the state variable method?
Answer: The state variable analysis method not only discusses the input, but also considers the changes in the internal nodes of the system, lists the system state equation and response equation, and uses the knowledge of the matrix to solve the response.
4. What are the specific system analysis methods in the input-output method?
Answer: ① Time domain analysis: Through the initial conditions and differential equations of the system, and using methods such as homogeneous solutions, special solutions, convolution, etc., the system's full response, zero state response, etc. are obtained. The independent variable is the time domain. Advantages It intuitively reflects the signal changes over time, but the disadvantage is that the amount of calculation is large.
②Transform domain analysis: Including Fourier transform and Laplace transform, the signal is transformed from the time domain to the frequency domain and the complex frequency domain. The system function, system impulse response, convolution and inverse transformation can be obtained. The advantage of quickly finding the system response is that the amount of calculation is small, but the disadvantage is that the physical meaning is not as good as the time domain.
5. What is the representation form of a continuous system?
Answer: ① Differential equation
② Unit impulse response h (t)
③ Network function H (jw) - Fourier transform
④ System function H (s) - Laplace transform
6. How to judge the causality, stability and filtering characteristics of the system?
Answer:
(1) Causality:
① Definition: The output at the current moment only depends on the input at the current moment and the previous moment, and has nothing to do with the future.
②In the Laplace transform, if the convergence domain is on the right side of the straight line, there is cause and effect.
(2) Stability:
①Definition: If the input is a bounded signal, the output is also a bounded signal.
② In Laplace transform: If the convergence domain includes the imaginary axis, it is stable.
(3) Filter characteristics: You can draw the amplitude-frequency response curve through the transformation from time domain to frequency domain to see whether it is in the form of high pass, low pass, band pass, etc.
7. Tell us about the concept and function of Fourier transform?
Answer: The concept of Fourier transform: Fourier transform converts a signal function f(t) in the continuous time domain into a function F(ω) in the continuous frequency domain, where ω is the frequency. This transformation is achieved by decomposing the signal into a superposition of sine and cosine functions of different frequencies. My understanding is that a signal can be synthesized by superposing trigonometric function signals of various frequencies. The more triangular signals are superimposed, the closer the synthesized signal will be to the original signal.
I think the main functions of Fourier transform are:
①Spectral analysis: Fourier transform can decompose a signal into components of different frequencies, thus providing information about the signal in the frequency domain, allowing us to perform spectrum analysis. By analyzing the spectrum of a signal, we can understand the various frequency components contained in the signal, such as harmonics, noise, etc.
②Signal processing: By converting the signal from the time domain to the frequency domain, we can perform operations such as filtering, noise reduction, and modulation on the signal to change the characteristics of the signal.
③Modulation and demodulation: In communication systems, FT can use different carrier frequencies to modulate the fundamental signal to different frequency bands to improve frequency band utilization. During demodulation, FT can be used to perform appropriate filtering to perfectly restore the original signal.
8. Why do we need Laplace transform and Z transform after learning Fourier transform?
Answer: First, a signal satisfies the absolute integrability condition, that is, when the integral of the signal is less than infinity (the signal can be considered to be attenuated), FT exists. Well, in fact, many signals do not meet the absolute integrability condition, such as polynomials, exponential functions, etc.
The Laplace transform transforms these time-continuous signals that do not satisfy the FT condition from the time domain to the frequency domain. For example, y=x^3, its growth rate is greater than the growth rate of the complex exponential signal, so it cannot be fitted with a complex exponential signal, but we can multiply it by an attenuation factor so that this signal can match the conditions of Fourier transform.
Z transform is to transform these time discrete signals that do not meet FT conditions from the time domain to the frequency domain.
9. Tell me what you understand about the concepts of Laplace transform and convergence region?
Answer: The Laplace transform is to transform these time-continuous signals that do not meet the FT conditions from the time domain to the frequency domain. For example, y=x^3, its growth rate is greater than the growth rate of the complex exponential signal, so it cannot be fitted with a complex exponential signal, but we can multiply it by an attenuation factor so that this signal can match the conditions of Fourier transform.
The convergence region is actually the value of δ in the attenuation factor e^(δt) that allows the signal to meet the conditions of Fourier transform after convergence. The convergence domain of Laplace transform corresponds to rectangular coordinates and is a plane.
10. Tell me about your understanding of Z transformation and its convergence region?
Answer: Z transform is for digital signals. This type of discrete signal is also similar to Fourier transform. It is decomposed into discrete complex exponential signals of different frequencies, and the corresponding coordinates are obtained. For signals that do not meet the FT condition, multiply the attenuation factor so that the discrete complex exponential can keep up with the transformation of the signal.
The convergence region of the Z-ratio transformation is the unit circle in polar coordinates.
11.What are the methods for solving zero-state response? What are full response, free response, and forced response?
Answer: ① Direct solution: According to the differential equation or difference equation of the system, use the initial conditions and input signals to directly solve the zero state response.
② Splitting method (separation of variables method): For linear time-invariant systems, the input signal can be decomposed into a linear combination of basic unit responses (such as impulse response or step response). The zero-state response is equal to the convolution of the excitation signal and the system impulse response.
③ Laplace transform: Use Laplace transform to convert differential equations into algebraic equations. By solving algebraic equations, the system function H(s) of the system can be obtained. Then the transfer function is multiplied by the Laplace transform of the input signal and then inversely transformed to obtain a zero-state response.
④Z transform: For discrete-time systems, Z transform can be used to convert the difference equation into an algebraic equation. By solving the algebraic equations, the transfer function of the system can be obtained. Then the transfer function is multiplied by the Z transform of the input signal, and then the inverse transform is performed to obtain a zero-state response.
12. What are the spectrum characteristics of periodic signals?
Answer: ① Discreteness: The independent variable of the spectrum is discontinuous and discrete.
②Harmonicity: There are many harmonics in the spectrum, and harmonics only appear at integer multiples of the fundamental frequency Ω.
③Convergence: As the harmonic order increases, the amplitude of the harmonics will gradually decrease.
13.What is Pasval’s theorem?
Answer: Pasval's theorem is a mathematical theorem that describes the energy conservation relationship between the signal in the time domain and the frequency domain. Simply put, Pasval's theorem tells us that the energy of a signal is equal in time domain and frequency domain representation.
In the continuous time case, we can represent the signal as a function x(t) and convert it into a frequency domain representation X(f) through the Fourier transform. Pasval's theorem tells us that in time domain representation, the energy of a signal can be found by calculating the square of the amplitude of the signal at each time point and integrating it. In the frequency domain representation, the energy of the signal can be obtained by calculating the square of the amplitude of the signal at each frequency point and integrating it. Pasval's theorem states that these two energy values are equal.
14. Why does the sound become sharper when the radio speeds up?
Answer: This is related to the relationship between the time domain and frequency domain of the signal. When the time domain is compressed, the corresponding frequency domain will be broadened. The spectrum of the signal is composed of fundamental wave and high-frequency harmonic components. When the frequency domain is broadened, the original filter cannot filter out the new high-frequency components, causing the sound signal amplitude to become higher.
15. What are the conditions for distortion-free transmission of the system?
Answer: ① The amplitude-frequency characteristic of the system is a straight line parallel to the x-axis, which is a constant.
②The phase-frequency characteristic of the system is a straight line passing through the origin.
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"Digital Signal Processing"
1.Definition of digital signal processing
Process the digital sequence and transform the signal into a form that meets the requirements. It mainly includes two parts: digital spectrum analysis and digital filtering.
Chapter 2 Time Domain Discrete Time Signals and Systems
1. Sampling theorem
The sampling theorem, also known as the Nyquist-Shannon sampling theorem, means that when sampling, in order to completely restore the original signal, the sampling frequency must meet certain conditions.
The sampling theorem is expressed as follows: If the highest frequency of a signal in the continuous time domain is f_max, then in order to completely recover the signal, the sampling frequency f_s must satisfy f_s > 2f_max.
In other words, the sampling frequency must be greater than twice the highest frequency of the signal. This is because according to the Nyquist theorem, when sampling, the spectrum of the signal will be copied infinitely to a frequency range that is half of the sampling frequency. If the sampling frequency is less than twice the highest frequency of the signal, then these spectral copies will overlap, resulting in the sampled signal not being able to fully recover the original signal.
2. Consistent with the content of "Signals and Systems":
① Some common discrete digital sequences
② Operations such as addition, multiplication, deconvolution, displacement, convolution and sum of sequences
③ Linearity (homogeneity and additivity) of linear non-shift-variant systems (i.e. linear non-time-variant systems) ), time-invariance, the relationship between unit impulse response and convolution sum
④ Properties of LTI system: commutative law, associative law, distributive law
⑤ Judgment of stable system: bounded input and bounded output
⑥ Causal system: systematic The output only depends on the moment of input and before that moment, and has nothing to do with the future.
⑦Linear constant coefficient difference equation: Similar to the differential equation of continuous signals, the difference equation is suitable for discrete time systems.
Chapter 3 Frequency Domain Analysis of Discrete Time Signals and Systems
1. Continuous time signals have Fourier transform, and discrete time series also have Fourier transform.
2. Z transform: including the definition, properties, convergence domain, inverse transform, and basic theorem of Z transform
3. The relationship between Fourier, Laplace and Z transform
Chapter 4 Discrete Fourier Transform
Background: The Fourier transform of a sequence is a discrete transform in the time domain but a continuous transform in the frequency domain, which is not conducive to computer processing. Now we want to study an important finite-length sequence, and invented the DFT (discrete Fourier transform) to realize the simultaneous discretization of the time domain and frequency domain.
1. Several forms of Fourier transform
①Continuous aperiodic signal: the frequency domain is also continuous and aperiodic ②Continuous
periodic signal: the frequency domain is aperiodic and discrete ③Discrete
aperiodic signal: the frequency domain is periodic and continuous
④Discrete periodic signal: the frequency domain is periodic and discrete of
One characteristic can be seen: the continuity and periodicity of the time domain and frequency domain are relative and opposite. *
Time domain - frequency domain
Periodic - discrete
Aperiodic - continuous
Continuous - aperiodic
Discontinuous - periodic
2. Discrete Fourier Series——DFS
In fact, it is not important. The concept of periodic convolution is introduced here.
What is the difference between periodic convolution (circular convolution) and linear convolution (finite-length sequence convolution)?
①Periodic convolution is the convolution of two sequences with a period of N, and the result is also a periodic sequence. Linear convolution targets finite-length sequences, and the result is also finite-length.
② Periodic convolution is only performed on one period during deconvolution and summation. The summation of linear convolution is performed over the entire sequence.
③The result of linear convolution after periodic extension is equivalent to the periodic convolution after periodic extension of these two finite-length sequences.
3. Discrete Fourier Transform - DFT
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"Digital Signal Processing?" some interview questions
1. What is sampling? What is quantification? What is encoding?
Answer: Sampling refers to the process of converting continuous time signals into discrete time . During the sampling process, continuous-time signals are measured or collected within a certain time interval to obtain discrete-time sample values . The sampling theorem (Nyquist theorem) states that in order to completely restore the original signal, the sampling frequency must be greater than or equal to twice the highest frequency in the original signal.
Quantization refers to the process of converting a continuous amplitude signal into discrete amplitudes . In the quantization process, the sample values of the *sampled continuous-time signal are mapped to discrete amplitude levels . During the quantization process, the amplitude of the signal is divided into a finite number of discrete levels, each level corresponding to a discrete amplitude value. The purpose of quantization is to convert the infinite possible amplitude values of a continuous signal into finite discrete amplitude levels for digital representation and processing. Quantization error or quantization noise may occur .
Encoding refers to the process of converting signals of discrete time and discrete amplitude into digital form. During the encoding process, the quantized discrete amplitude values are represented as binary codes (or other digital encoding forms) to facilitate the storage, transmission, and processing of digital signals . During the encoding process, each discrete amplitude value is mapped to a corresponding binary code or encoding form. The purpose of encoding is to represent discrete amplitude values in digital form so that the signal can be processed by a computer or other digital device.
2. What is the process of digital processing of analog signals?
Answer: ① Pre-filter processing - to prevent spectrum aliasing distortion caused by sampling.
② Value acquisition and quantization in AD converter
③ Digital processing
④ DA converter to restore analog signal
⑤ Post filter - filter out high-frequency noise and increase accuracy
3. Is a sinusoidal sequence necessarily a periodic sequence?
Answer: Not necessarily. For sin wt, if 2Π/w is a rational number, it is a periodic sequence. If it is an irrational number, it is not a periodic sequence.
4. What is the difference between DSP and FPGA?
A DSP is a specialized microprocessor designed to process digital signals efficiently. DSP chips usually have specialized hardware instruction sets and optimized algorithms to achieve high-speed signal processing in real-time applications. DSP chips usually have higher clock frequencies and parallel computing capabilities, and are suitable for audio and video processing, communication systems, radar signal processing and other fields. DSP generally has higher power consumption, but has higher efficiency and flexibility in processing digital signals.
FPGA is a programmable logic chip, which consists of a large number of programmable logic gates and programmable connections. FPGA can perform flexible hardware logic configuration according to the designer's needs. FPGA can be programmed to implement different circuit functions and is widely used in applications that require rapid prototyping, low power consumption, customized circuits and highly parallel computing. FPGAs typically have lower clock frequencies and higher power efficiency, but have higher flexibility and customizability in terms of programmable logic and parallel computing.
DSP and FPGA have some overlap in application fields and design methods, but there are also some differences. DSP is more suitable for applications such as digital signal processing, real-time control and algorithm optimization, while FPGA is more suitable for applications such as customized circuits, rapid prototype development and highly parallel computing. The choice of using a DSP or FPGA depends on the specific application needs, performance requirements and design complexity.
5.What are the common basic operations in DSP systems?
Answer: Adders, multipliers, displacement operations
6. Tell us about the Nyquist theorem?
Answer: The Nyquist theorem is mainly used when sampling analog signals. Only when the sampling frequency is at least twice the highest frequency of the signal can the frequency spectrum be aliased and information will not be lost. Below 2 times, spectrum aliasing will occur.
7. What is spectral aliasing? How to avoid spectral aliasing?
Answer: When the sampling frequency does not satisfy the Nyquist theorem, the high-frequency signal frequency will be incorrectly mapped to the low-frequency frequency. This is spectrum aliasing. The aliased signal spectrum overlaps with the original signal spectrum, making it impossible to accurately restore the original signal.
Possible: Use an anti-aliasing filter: Use an anti-aliasing filter to preprocess the signal before sampling it. The anti-aliasing filter removes frequency components above half the sampling frequency from the signal before sampling to avoid spectral aliasing.
8. What is the correspondence between continuity and periodicity in the time domain and frequency domain?
Answer: ①Continuous aperiodic signal: the frequency domain is also continuous and aperiodic
②Continuous periodic signal: the frequency domain is aperiodic and discrete ③Discrete
aperiodic signal: the frequency domain is periodic and continuous
④Discrete periodic signal: the frequency domain is periodic ,dispersed
One characteristic can be seen: the continuity and periodicity of the time domain and frequency domain are relative and opposite. *
Time domain - frequency domain
Periodic - discrete
Aperiodic - continuous
Continuous - aperiodic
Discontinuous - periodic
9.What is the difference between DTFT and DFT? ————DFT is obtained by sampling the frequency domain based on DTFT.
Answer: DTFT is the discrete time Fourier transform. Its time domain is discrete, but the frequency domain is continuous.
DFT is the discrete Fourier transform, and its time domain and frequency domain are both discrete.
I think this diagram is the best diagram to explain DFT and DTFT.
We know that when the signals are complex and large, human calculations are not as good as computer calculations. However, computers can only process digital signals, so we must convert analog signals into digital signals.
First, let’s talk about Figures (1) and (2). For an analog signal, as shown in Figure (1), to analyze its frequency components, it must be transformed into the frequency domain. This is done through Fourier Transform (FT) Obtained, then we have the spectrum of the analog signal, as shown in Figure (2); Note 1: Both the time domain and frequency domain are continuous!
However, computers can only process digital signals. First, the original analog signal needs to be discretized in the time domain, that is, sampled in the time domain. We find a sampling pulse sequence as shown in Figure (3), and the spectrum of this sampling sequence is as shown in Figure (4). It can be seen that its spectrum is also a series of pulses .
Multiply the analog signal and the sampling sequence in the time domain. After (1)×(3), the discrete time signal x[n] can be obtained, as shown in Figure (5); the multiplication in the
time domain is equivalent to the volume of the frequency domain. product, then in the frequency domain, if Figure (2) is convolved with Figure (4), a mirror image will appear at each pulse point, so Figure (6) is obtained, which is the discrete time signal x[n shown in Figure (5) ] DTFT, the discrete time Fourier transform
The emphasis here is on the four words "discrete time". Note that the time domain is discrete, while the frequency domain is still continuous.
After the above two steps, the signal we get still cannot be processed by a computer because the frequency domain is both continuous and periodic. We naturally think that since the time domain can be sampled, why can't the frequency domain be sampled ? Doesn’t this discretize both the time domain and the frequency domain?
That's right, next the frequency domain is sampled. The spectrum of the frequency domain sampling signal is shown in Figure (8), and its time domain waveform is shown in Figure (7).
Now we perform frequency domain sampling, that is, frequency domain multiplication. Figure (6 ) Figure (7) is convolved to obtain figure (9). As expected, the mirror image will appear periodically at each pulse point. We take the main value interval of the periodic sequence in Figure (10) and mark it as X(k), which is the DFT of the sequence x[n], that is, the discrete Fourier transform . It can be seen that DFT only samples DTFT in the frequency domain and intercepts the main values for the convenience of computer processing . Some people may be confused. If you perform IDFT on Figure (10) and return to the time domain, which is Figure (9), it is different from x[n] shown in the original discrete signal diagram (5). It is the periodicity of x[n]. Extend! That's right, so if you look up the definition of IDFT, does it limit the value range of n? The meaning of this restriction is that x[n] can be restored by taking the main value interval of the periodic extension sequence!
What about FFT? FFT was proposed solely for the purpose of quickly calculating DFT. Its essence is DFT! The algorithms included in our commonly used signal processing software MATLAB or DSP software packages are FFT rather than DFT.
DFS is proposed for time domain periodic signals. If you perform DFS on the periodic extension signal shown in Figure (9), you will get Figure (10). As long as its main value interval is intercepted, it has a complete one-to-one correspondence with DFT. precise relationship. This can be easily seen by comparing the definitions of DFS and DFT. Therefore, the essence of DFS and DFT are the same, but they are described in different ways.
10.What are the conditions for the existence of DTFT?
Answer: Similar to FT, FT satisfies absolute integrability, and DTFT satisfies absolute summability.
11.What is the relationship between DTFT and Z transform?
Answer: DTFT is a special case of Z transform on the unit circle. When the complex variable z of the Z transform is on the unit circle, that is, |z|=1, the Z transform becomes DTFT. On the unit circle, the frequency response of the Z transform is equal to the frequency response of the DTFT.
Only when the convergence domain of the Z transform of the sequence includes the unit circle, the conditions for the existence of DTFT are met.
Therefore, it can be said that DTFT is a special case of Z transform on the unit circle. DTFT describes the frequency characteristics of discrete time signals in the frequency domain, while Z transform describes the relationship between the input and output of a discrete time system in the complex variable domain.
12.What are the disadvantages of DTFT? Why use DFT?
Answer: If the input sequence is an infinitely long sequence, each spectrum will be calculated by infinite additions and multiplications when using DTFT analysis. This means that the amount of calculation is too large and there is not enough capacity to store data. Therefore, for infinitely long sequences, it is basically not possible. Possible implementation of DTFT.
When we perform DFT processing, we only calculate the DFT of N sample values within one frequency cycle, so that the frequency domain of the signal is finite and discretized, thereby increasing the speed of signal processing.
12.5 Since DFT also requires sampling in the frequency domain, what is the sampling theorem in the frequency domain?
Answer: Frequency domain sampling will cause periodic extension in the time domain, so only when the number of sampling points N is greater than the sequence length M, the sampled spectrum can restore the original sequence without distortion.
13. What are the differences between linear convolution (sum of convolutions), N-point circular convolution (cyclic convolution), and N-point periodic convolution?
Answer:
(1) Linear convolution, also called convolution sum, is aimed at two aperiodic sequences with different lengths. The length of the sequence after convolution is M+N-1.
(2) Circular convolution (circular convolution) is aimed at two sequences with different lengths. If the lengths are different, the shorter one needs to be padded with zeros until both are N points. Then perform periodic extension to turn the finite-length sequence into a periodic sequence, and then perform defolding, taking the main value interval, and shift addition. Note: The convolution of a finite-length sequence after periodic extension is called periodic convolution, and the circular convolution is that the result of periodic convolution takes the main value interval, so the result is a finite-length non-periodic sequence.
(3) Periodic convolution: The convolution of a finite-length sequence through periodic extension is called periodic convolution, and the result is also a periodic sequence.
14. Why is Fast Fourier Transform FFT introduced?
Answer: (1) First of all, DFT has a large amount of operations. Calculating N times requires N^2 complex multiplications and N (N-1) complex additions.
The time complexity of the traditional DFT algorithm is O(N^2) , where N is the length of the input sequence. This means that when the input sequence length is large, the computational overhead of DFT is very large, limiting real-time and efficient signal processing applications.
(2) FFT is a fast algorithm for calculating DFT based on the idea of divide-and-conquer. It greatly reduces the computational complexity by splitting the calculation of DFT into multiple smaller sub-problems and taking advantage of the characteristics of symmetry and periodicity. The time complexity of the FFT algorithm is O(NlogN) , which is much faster than the traditional DFT algorithm. It is especially suitable for large-scale signal processing and spectrum analysis.
15. What is the principle of time-based DIT-FFT?
Answer: Time-based FFT divides a long sequence into two short sequences according to odd and even.
The principle of time-based FFT is as follows:
Input signal: Given a discrete time series x(n) of length N, where 0 ≤ n < N.
Decomposition step: Decompose the input sequence x(n) into two subsequences with even indexes and odd indexes. Let x_e(n) represent the even-indexed subsequence and x_o(n) represent the odd-indexed subsequence.
Recursive operation: perform recursive time-based FFT operations on the subsequences x_e(n) and x_o(n) respectively. The termination condition of the recursion is that the length of the subsequence is 1, in which case the subsequence is returned directly.
Fusion step: merge the results obtained by the recursive operation into the overall FFT result. For each k (0 ≤ k < N/2), calculate the FFT result X(k) using the following formula:
X(k) = X_e(k) + W_N^k * X_o(k)
Among them, X_e(k) and X_o(k) are the FFT results of the subsequences x_e(n) and x_o(n) respectively, W_N^k is the rotation factor, and the calculation formula is:
W_N^k = e^(-j * 2π * k / N)
Rearrange the results: Rearrange the FFT results in the correct order so that the frequencies go from 0 to N-1.
Taking the sequence of N=8 as an example, decompose it into two 4-point DFTs, and then perform 4 butterfly operations to find the values of all 8-point DFTs.
16.What are the number of complex multiplications and complex additions of FFT?
Answer: The number of complex multiplications of FFT: N/2 ✖logN and the number of complex additions: N logN
The number of complex multiplications of DFT: N^2 and the number of complex additions: N (N-1)
17.How does FFT simplify the computational complexity of DFT?
Answer: The base-2 FFT refers to continuously decomposing the DFT sequence of N points into DFTs with smaller points. The symmetry and periodicity of the rotation factor are used to simplify the calculation. When N is the m power of 2, m can be drawn Sub-butterfly diagram, recursive operation. Parity grouping in the time domain and before and after grouping in the frequency domain decompose long sequence DFT calculations into short sequence DFT calculations.
18. Digital filters are divided into IIR (infinite impulse response type) and FIR (finite impulse response type). How about introducing these two types?
IIR system: The unit impulse response is an infinite-length sequence. Since the system function has both zeros and poles, it must be a recursive structure.
FIR system: The unit impulse response is finite and non-recursive.
19.What are the design steps of IIR filter?
Answer: 1: Determine the passband cutoff frequency, stopband cutoff frequency, 3dB cutoff frequency (amplitude reduced to square root 2/2, etc.) of the digital filter, and select a suitable design indicator for the analog filter.
2. Normalize the prototype analog filter, and set the passband cutoff frequency to 1.
Three: Use the bilinear transformation method or the impulse response invariant method to convert the analog filter into a digital filter.
20. The principles of bilinear transformation method and impulse response invariant method and their advantages and disadvantages.
Answer:
The bilinear transformation method is a single-value mapping from the S plane to the Z plane. The non-bandlimited analog filter is nonlinearly mapped into a bandlimited analog filter with the highest angular frequency π/T . The advantage is that it does not produce spectral aliasing, but the disadvantage is that it causes amplitude distortion due to nonlinear mapping .
The impulse response invariant method uses the idea of time domain approximation and uses the unit sampling response of the digital filter to imitate the unit sampling response of the analog filter . The advantage is that the designed digital filter has a linear phase and does not produce distortion. The disadvantage is that if the frequency response of the analog filter is not a band-limited signal, the frequency response of the designed digital filter will have aliasing and can only be used to design band-limited analog filters.
21. Butterworth filter and Chebyshev filter?
Answer: Butterworth filter: The flattest in the passband, but the frequency response in the transition band and stopband is relatively slow.
Chebyshev filter: It has equal ripple characteristics in the passband, that is, ripples appear in the passband, but it has high suppression ability in the transition band and stopband.