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Chapter 2. Signals

Chapter 2. Signals. Husheng Li The University of Tennessee. Homework 2. Deadline: Sept. 16, 2013. Spectrum. Physically, the signal is transmitted in the time domain. It is more convenient to study the signal in the frequency domain. The frequency domain description is called the spectrum.

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Chapter 2. Signals

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  1. Chapter 2. Signals Husheng Li The University of Tennessee

  2. Homework 2 Deadline: Sept. 16, 2013

  3. Spectrum Physically, the signal is transmitted in the time domain. It is more convenient to study the signal in the frequency domain. The frequency domain description is called the spectrum. The frequency description of signal can be obtained from Fourier transform:

  4. Example: Rectangular Pulse Time domain Frequency domain

  5. Signal Energy Rayleigh’s Theorem: The signal energy is given by Integrating the square of the amplitude spectrum over all frequency yields the total energy. |V(f)|^2 is called the energy spectral density.

  6. Band Limited Signals A signal should not use all bandwidth. Hence, we have to limit its band. Sinc function is a band limited one A band limited signal is infinite in the time, which is impossible in practice.

  7. Frequency Translation We need to transform a baseband signal to much higher frequency one. (Why?) It is equivalent to multiplying a sinusoidal signal having the carrier frequency.

  8. RF Pulse time frequency

  9. Convolution When a signal is passed through a linear time invariant (LTI) system, the output is the convolution of the input signal and the system impulse response. In the frequency domain, the convolution is equivalent to multiplication:

  10. Transfer Function Each LTI system can be represented by its transfer function.

  11. Signal Transmission: Distortionless Case The output is undistorted if it differs from the input only by a multiplying constant and a finite time delay: In the frequency domain, it is equivalent to In practice, the signal is always distorted.

  12. Linear Distortion: Amplitude High frequency attenuated Low frequency attenuated Linear distortion includes any amplitude or delay distortion associated with a linear transmission system, which is easily descried in the frequency domain. The amplitude could be distorted.

  13. Linear Distortion: Phase If the phase shift is not linear, the various frequency components suffer different amounts of time delay, called phase or delay distortion. The delay is given by

  14. Two Waveforms: Example

  15. Equalization Digital transversal filter Linea distortion is theoretically curable through the use of equalization networks.

  16. Multipath in Wireless The multiple paths in wireless communications cause different delays along different paths, thus causing inter-symbol interference. For example, consider two paths:

  17. Destructive Interference (two-path)

  18. Nonlinear Distortion Many devices could have nonlinear transfer characteristics. The nonlinear transfer characteristic may arouse harmonics.

  19. Transmission Loss Power gain: g=P_out / P_in dB scale: g_dB = 10 log_10 g For linear system of communication channel, we have

  20. Typical Values of Power Loss

  21. Example: Radio Transmission For the case of free-space transmission, the loss is given by Consider the antenna gains, the received power is given by

  22. Example: Satellite Communication

  23. Doppler Shift A passing automobile’s horn will appear to change pitch as it passes by. The change in frequency is called Doppler shift. When the moving speed is v and the angle is ϕ, the Dopper shift is

  24. Homework Deadline: Sept. 9, 2013

  25. Ideal Filter An ideal bandpassfilter is given by

  26. Filtering Perfect bandlimitiing and timelimiting are mutually incompatible. Rise time is a measure of the ‘speed’ of a step response:

  27. Quadrature Filter A quadrature filter is an allpass network that merely shifts the phase of the positive frequency components by -90 degrees. The output of a quadrature filter is called the Hilbert transform of the input.

  28. Properties of Hilbert Transform

  29. Bandpass Signals and Systems A bandpass signal has the following frequency domain property: The time domain bandpass signal can be written as

  30. Spectrum and Waveform of Bandpass Signal

  31. Quadrature-Carrier Description of Bandpass Signal A bandpass signal can be decomposed to in-phase and quadrature components:

  32. Frequency Domain of Bandpass Signal The frequency domain of a bandpass signal is given by The in-phase and quadrature functions must be lowpass signals:

  33. Lowpass Equivalent Signal In the frequency domain, we have the low pass equivalent spectrum: In the time domain, we have the lowpass equivalent signal: In the frequency domain, we have

  34. Lowpass-to-bandpass transformation The connection between and is given by In the frequency domain, we have

  35. Bandpass Transmission We can work on the lowpass equivalent spectra directly:

  36. Carrier and Envelop Delay If the phase shift is nonlinear, we can approximate it by using the Taylor’s expansion:

  37. Bandwidth and Carrier Frequency A large bandwidth requires high carrier frequency.

  38. Bandwidth: Definition Absolute bandwidth 3 dB bandwidth Noise equivalent bandwidth Null-to-null bandwidth Occupied bandwidth Relative power spectrum bandwidth

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