ELEC3002-complex5

# ELEC3002-complex5 - ELEC3002/ELEC7005 Lecture 5 An...

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ITEE Complex 5/1 ELEC3002/ELEC7005 ELEC3002/ELEC7005 Lecture 5 An application using complex analysis: The Time-Frequency Domain

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ITEE Complex 5/2 Time-frequency domain. What is it? Time-frequency domain. What is it? We are well acquainted with both time and frequency representations of a signal. Let’s consider a few examples first. Consider first a signal with constant amplitude and with linear increasing frequency. This is called a chirp. Also used extensively in radar for pulse compression
ITEE Complex 5/3 Consider the energy spectrum of the Consider the energy spectrum of the chirp chirp We cannot say anything from this plot about the evolution in time of the frequency content. This is due to the fact that the Fourier transform is a decomposition of exponentials which are of infinite duration and un-localized in time.

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ITEE Complex 5/4 time and frequency simultaneously? time and frequency simultaneously? It would be better to represent the signal in both the time and frequency domains simultaneously. This is precisely the aim of the time-frequency domain. This is called the Wigner-Ville distribution of the signal. The linear progression with frequency is clearly shown.
ITEE Complex 5/5 Noisy signal Noisy signal Now add some noise, first to the chirp time domain energy spectrum

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ITEE Complex 5/6 now for the time-frequency plot. now for the time-frequency plot. The Wigner distribution still shows the linear increase in frequency.
ITEE Complex 5/7 Another example Another example A sonar signal from a bat, recorded with a sampling frequency of 230.4kHz and an effective band width of 8-80kHz.

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ITEE Complex 5/8 Energy spectrum of the bat signal Energy spectrum of the bat signal frequency is isolated around 38-40kHz, but we do not know how this spectrum evolves in time.
ITEE Complex 5/9 In the time-frequency domain In the time-frequency domain This is the Pseudo-Wigner Ville distribution of the bat signal. narrow band signal, frequency content decreasing from around 48kHz to 38kHz with a nonlinear frequency modulation of approx hyperbolic shape.

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ITEE Complex 5/10 Fourier Transform Fourier Transform The time and frequency domains are linked via the Fourier Transform ( 29 2 1 ( ) 2 j t X x t e dt πϖ ϖ π - - = The spectrum tells us what frequencies are contained in the signal as well as corresponding amplitudes and phases, but not at what
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ELEC3002-complex5 - ELEC3002/ELEC7005 Lecture 5 An...

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