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В конспекте лекций представлен материал, читаемый в рамках курса «Основы цифровой обработки сигналов». В конспекте отражён базовый материал, необходимый для освоения данного курса и для подготовки к практическим работам, семинарам, зачётам и экзаменам. Конспект лекций предназначен для магистров направлений 11.04.02 «Инфокоммуникационные технологии и системы связи» и 11.04.04 «Электроника и наноэлектроника».

• Chapter 1 Basic knowledge
  • §1.1 Geometric progression and series
  • §1.2 Complex numbers
  • §1.3 Trigonometric expressions
    • 1.3.1 Basic formulas
    • 1.3.2 Integrals
    • 1.3.3 Orthogonality
  • §1.4 Linear operators
  • §1.5 Convolution
    • 1.5.1 Linear convolution
    • 1.5.2 Cyclic convolution
  • §1.6 Fourier series
  • §1.7 Integral Fourier Transform
    • 1.7.1 Definition
    • 1.7.2 Spectrum of signal
    • 1.7.3 Properties
    • 1.7.4 Sine and cosine transforms
    • 1.7.5 Shifting theorem
    • 1.7.6 Theorem of convolution
    • 1.7.7 General formulas
  • §1.8 Laplace Transform
    • 1.8.1 Definition and properties
    • 1.8.2 Impulse response and transfer function
    • 1.8.3 Poles and stability
  • §1.9 Z-transform
    • 1.9.1 Definition
    • 1.9.2 Connection with other transforms
  • §1.10 Dirac delta function
• Chapter 2 Discrete sequences and systems
  • §2.1 Introduction
  • §2.2 Operations on discrete sequences
  • §2.3 Real-time systems
  • §2.4 Complexity metrics
  • §2.5 Unit delay element
  • §2.6 Discrete linear systems
  • §2.7 Time-invariance systems
• Chapter 3 Sampling signals
  • §3.1 Ambiguity of signal presentation
  • §3.2 Discrete-Time Fourier transform
    • Home exercise: check that spectrum is periodic.
  • §3.3 Discrete sequence spectrum
  • §3.4 Signal reconstruction
    • Home exercise: get spectrum of the ideal DAC.
  • §3.5 Sampling low-pass signals
  • §3.6 Sampling band-pass signals
    • 3.6.1 Limits for band-pass sampling
    • 3.6.2 Inversion
    • 3.6.3 Recommendations
    • Home exercise: prove the statement above.
• Chapter 4 Discrete Fourier Transform
  • §4.1 Derivation
  • §4.2 DFT example
  • §4.3 Properties of DFT
    • 4.3.1 Axes conversion (magnitude and frequency)
    • 4.3.2 How T, fs and N effect on spectrum?
    • 4.3.3 Linearity
    • Home exercises: proof linearity.
    • 4.3.4 Shifting theorem
    • 4.3.5 Theorem of convolution
    • Home exercise: proof theorem of convolution.
    • 4.3.6 Symmetry
  • §4.4 Symmetric DFT forms
  • §4.5 DFT matrix
  • §4.6 DFT of typical functions
    • 4.6.1 General rectangular function
    • 4.6.2 Symmetric rectangular function
    • 4.6.3 Constant level
    • 4.6.4 IDFT of rectangular function
    • 4.6.5 Complex signal
    • 4.6.6 Real signal
  • §4.7 Leakage
  • §4.8 Windows
  • §4.9 Signal to noise ratio in DFT
  • §4.10 Conclusion
• Chapter 5 Fast Fourier Transform
  • §5.1 Algorithm
    • 5.1.1 Derivation
    • 5.1.2 Illustration of calculation flow
    • 5.1.3 Complexity of calculation
  • §5.2 Bit-reversed order
  • §5.3 Butterfly structures