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Concept of frequency in Discrete Signals & Introduction to LTI Systems

Concept of frequency in Discrete Signals & Introduction to LTI Systems. Concept of frequency in Discrete Signals. Concept of frequency in Discrete Signals. Digital Filters. Digital Filters. Fourier Series for continuous time periodic signals.

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Concept of frequency in Discrete Signals & Introduction to LTI Systems

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  1. Concept of frequency in Discrete Signals & Introduction to LTI Systems

  2. Concept of frequency in Discrete Signals

  3. Concept of frequency in Discrete Signals

  4. Digital Filters

  5. Digital Filters

  6. Fourier Series for continuous time periodic signals

  7. Fourier Transform Theorem & PropertiesReview of CTFT Frequency domain representation of a continuous-time signal The continuous-time signal xa(t) can be recovered from it’s CTFT, Xa(jΩ) we denote the CTFT pair as

  8. Fourier Series for discrete time periodic signals

  9. Fourier Transform Theorem & PropertiesDiscrete-Time Fourier Transform Representation of a sequence in terms of complex exponential sequence, {ejωn} The DTFT pair,

  10. Introduction to LTI System Discrete-time system x[n] Input sequence y[n] Output sequence Fig: Example of a single-input, single-output system • Discrete-time Systems • Function: to process a given input sequence to generate an output sequence

  11. Introduction to LTI System Linear System Most widely used A Discrete-time system is a linear system if the superposition principle always hold. If y1[n] and y2[n] are the response to the input sequences x1[n] and x2[n], then Linear DTS x[n] = αx1[n] + βx2[n] y[n] = αy1[n] + βy2[n]

  12. Example Is the system described below linear or not ? y[n] = x[n] + x[n-1] Step : a. Now, applying superposition by considering input as : x[n] = ax[n] + bx[n] b. Substitute the equation above with equation in (a), become y[n] = (ax[n] + bx[n]) + (ax[n-1]+ bx[n-1]) c. Rearrange the equation above become :- y[n] = a(x[n] + x[n-1]) +b(x[n] + x[n-1]) => ay[n] + by[n] c. The system is Linear since superposition is hold. Introduction to LTI System

  13. Shift-invariant System/Time-Invariant System A shift (delay) in the input sequence cause a shift (shift) to the output sequence If y1[n] is the response to an input x1[n], then the response to an input x[n] = x1[n - no] is y[n] = y1[n - no] Introduction to LTI System

  14. Causal System Changes in output samples do not precede changes in input samples y[no] depends only on x[n] for n ≤ no Example: y[n] = x[n]-x[n-1] Introduction to LTI System

  15. Stable System For every bounded input, the output is also bounded (BIBO) Is the y[n] is the response to x[n], and if |x[n]| < Bx for all value of n then |y[n]| < By for all value of n Where Bx andBy are finite positive constant Introduction to LTI System

  16. If the input to the DTS system is Unit Impulse (δ[n]), then output of the system will be Impulse Response (h[n]). If the input to the DTS system is Unit Step (μ[n]), then output of the system will be Step Response (s[n]). Introduction to LTI System Impulse and Step Response

  17. A Linear time-invariant system satisfied both the linearity and time invariance properties. An LTI discrete-time system is characterized by its impulse response Example: x[n] = 0.5δ[n+2] + 1.5δ[n-1] - δ[n-4] will result in y[n] = 0.5h[n+2] + 1.5h[n-1] - h[n-4] Introduction to LTI System Input-output Relationship

  18. Introduction to LTI System Input-output Relationship x[n] can be expressed in the form where x[k] denotes the kth sample of sequence {x[n]} The response to the LTI system is or represented as

  19. Introduction to LTI System Input-output Relationship Properties of convolution Commutative Associative Distributive

  20. Causality Properties of LTI Systems

  21. Properties of LTI Systems Stability if and only if, sum of magnitude of Impulse Response, h[n] is finite

  22. Stability

  23. Properties of LTI Systems

  24. Properties of LTI Systems

  25. Properties of LTI Systems

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