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Cellular COMMUNICATIONS

Cellular COMMUNICATIONS. DSP Intro. Signals: quantization and sampling. Signals are everywhere. Encode speech signal (audio compression) Transfer encode signals using RF signal (modulation) Detect antenna signal

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Cellular COMMUNICATIONS

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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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