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Published byElla Rose Modified over 4 years ago

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Continuous Time Signals All signals in nature are in continuous time

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From Discrete Time to Continuous Time A continuous time signals can be viewed as the limit of a discrete time signal with sampling interval

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From Discrete Time FT (DTFT) … We saw the DTFT of a discrete time signal Substitute and obtain:

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… to Continuous Time FT Now take the limit so that discrete time -> cont. time Then we obtain the Fourier Transform sampling freq -> infinity sum -> integral

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Fourier Transform We want to represent a signal in terms of its frequency components. Define: Fourier Transform (FT)

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Example of a Fourier Transform Take a Rectangular Pulse

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Example of a Fourier Transform

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Properties of the FT: 1. Symmetry If the signal is real, then its FT is symmetric as since Example: just verify the previous example

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Symmetry of the FT Magnitude has “even” symmetry Phase has “odd” symmetry

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Properties of the FT: 2. Time Shift since In other words a time shift affects the phase, not the magnitude

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Bandwidth of a Baseband Signal A Baseband Signal has all frequency components at the low frequencies, around F=0 Hz; Bandwidth: the frequency interval where most of the frequency components are.

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What does it mean? If you take the signal at two different times and with then since

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For Example: zoom samples spaced by less than 0.1msec are fairly close to each other

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Computation of the Fourier Transform Whatever we do, physical signals are in continuous time and, as we have seen, they are described by the FT; The FT can be computed in one of two ways: 1.Analytical: if we have an expression of the signal (like in the example); 2.Numerical: by approximation using the Fast Fourier Transform (FFT).

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Fourier Transform and FFT Consider a signal of a finite duration with Bandwidth. Then we can approximate, by simple arguments, where (say at least an order of magnitude smaller)

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Fourier Transform and FFT Using the FFT: Take an even integer. Then compute the N point FFT of the sampled data, padded with zeros: Assign the frequencies: positive frequencies negative frequencies

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Example Take a sinusoid with frequency and length Let the sampling frequency be

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Example X=fft(x, N); F=(-N/2:N/2 -1)*Fs/N; plot(F,fftshift(20*log10(abs(X))))

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Example (Zoom in at the Peak) Max at F=10kHz Sidelobes due to finite data length

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Complex Signals All signals in nature are real. There is not such as a thing as “complex” signal. However in many cases we are interested in processing and transmitting “pairs” of signals. We can analyze them “as if” they were just one complex signal: Real Signals Complex Signal

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Amplitude Modulation: Real Signal You want to transmit a signal over a medium (air, water, space, cable…). You need to “modulate it” by a carrier frequency:

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Amplitude Modulation: Complex Signal However most of the times the signal we modulate is Complex Notice now that the modulated signal is real and it contains both signals a(t) and b(t).

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FT of Modulated Signal See the different steps:

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FT of Modulated Signal Put things together: Usually

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