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Digital Signal Processing Basics: Introduction and Applications

Understand signals, sampling, Fourier analysis, and digital systems in the context of cellular communications and DSP. Learn about quantization, frequency domains, Fourier series, sampling theorems, Nyquist frequency, filters, and LTI systems.

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Digital Signal Processing Basics: Introduction and Applications

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  1. Cellular COMMUNICATIONS DSP Intro

  2. Signals: quantization and sampling

  3. Signals are everywhere • Encode speech signal (audio compression) • Transfer encode signals using RF signal (modulation) • Detect antenna signal • Pack several calls into a single RF signal from the antenna (multiple access) • Improve faded signal (equalization) • Adjust transmitted signal power to save battery

  4. What is signal? • Continuous signal • Real valued-function of time x=x(t), t=0 is now, t<0 is the past • Can’t work with it in the computer • But easy to analyze • Discrete signal • A sequence s=s(n), n=0 is now • Values are quantized (e.g. 256 possible values) • Need a time scale: n=1 is 1ms, n=2 is 2 ms etc. • Can process by computer (finite portion a time)

  5. Discrete signal from continuous • Sampling • Sample value of a continuous signal every fixed time interval • Quantization • Represent the sampled value using fixed number of levels (N=255)

  6. Example:sampling

  7. Example

  8. Frequency Domain

  9. *almost* any wave from sine waves

  10. Frequency domain • Can decompose *almost* every signal into sum of sinusoids multiplied by a *weight* • Frequently domain=*weights* of sinusoids • Example: • Upper case letter for frequency domain • X(0)=0,X(1)=1,X(2)=0.4,X(3)=0 • X is the spectrum of x

  11. Example: Sawtooth Frequency Domain X(k)=1/k

  12. Spectrum of sawtooth

  13. Example: Box X(n)=1/n (n is odd), X(n)=0 (n is even)

  14. Spectrum of a linear combination • Spectrum of x1+x2 is • Spectrum of x1+ • Spectrum of x2

  15. Frequency Domain • *Almost* every good periodic function can be represented by • Two series (numbers) describe the function • Recall Taylor expansion (polynomial base) • Discreet Fourier Transform takes function and gives it’s Fourier representation • Inverse DFT….

  16. Representing Fourier Series • Coefficient of cosines and sinus • Cosine amplitude and phase • Still two series, not convenient

  17. DFT summary • Can go back and forth from time-domain to frequency domain representation • Can be computed efficiently (FFT) • Signal Power in frequency and time domain (Parseval theorem)

  18. Sampling theorem

  19. Periodic Sampling • Discrete signals are obtained from continuous signals (acoustic/speech, RF) by sampling magnitude every fixed time period • How much should sampling period be for obtaining a good idea about the signal • Too much samples: need more CPU, power, clock etc.

  20. Ambiguity problem

  21. Ambiguity • Sample Frequency: • Digital sequence representing also represent infinitely many other sinusoids

  22. Aliasing • Suppose our signal is composed of sinusoids from 1kHz to 4KHz (with varying weights) • At sampling rate of 5 kHz we can discard 1kHz+5kHz and 4+5kHz as we know that signal has only up to 5kHz • At sampling rate of 2kHz we can distinguish between 1kHz and 3kHz which both are possible

  23. Ambiguity in frequency domain

  24. Nyquist sampling frequency • Signal band • Avoid aliasing • Nyquist sampling frequency • Maximum frequency without aliasing

  25. Sampling low pass signals • A signal is within the known band of interest • But contains some noise with higher frequencies (above Nyquist frequency) • Spectrum of digital signal will be corrupted

  26. Low Pass Filter

  27. Time vs. Frequency Domain

  28. Spectrum of the pulse

  29. Time vs. Frequency • Short pulse in time domain->wide spectrum

  30. Power Spectral Density(PSD)

  31. PSD and Separation of signals

  32. Discrete systems

  33. Discrete System • Example:

  34. Operation with signals • Can add and subtract two signal • Graphical representation

  35. Summation

  36. Linear Systems • Simple but powerful • Easy to implement

  37. Example • Example 1Hz+3Hz sine waves

  38. Frequency domain vs. Time Domain • Analyze a discrete system in time domain • What it does to the sequence x(n) • Analyze a discrete system in frequency domain • What it does to the spectrum • Change in coefficient of various sinusoids of a signal

  39. Example:1Hz+3Hz

  40. Nonlinear Example: 1Hz+3Hz f(x1+x2)!=f(x1)+f(x2)

  41. Non-linear systems • Might introduce additional sinusoids not present in input • Results from interaction between input sinusoids • Difficult to analyze • Sometimes are used in practice • We stick to linear systems for a while

  42. Time-Invariant Systems • Has no absolute clock • Example:

  43. Example

  44. Unit Time Delay

  45. Time-Delay • Feasible system can’t look into a future • at n=0 can’t produce x’(0)=y(4) • only at n=4, can output x’(0)=y(4)

  46. LTI: Linear Time Invariant • LTI is easy to analyze and build. Will focus on them

  47. Analyzing LTI systems

  48. LTI systems • Linear • Time-Invariant • Recall linear algebra • A vector space has basis vectors • Linear operator completely defined by its behavior on basis vectors • LTI need to specify only on a single basis vector

  49. Vector Space of Signals • Shifted Unit Impulse(SUI) signal • Basis for representation of the digital signals

  50. SUI are a basis

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