Module-3-Lecture-Slides_9-10-2015

# Xt signoisefsnrn generate test signal where f

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[x,t] = sig_noise([f],[SNR],N); % Generate test signal where f specifies the frequency of the sinusoid(s) in Hz. If f is a vector, then a number of sinusoids at each frequency in f. SNR specifies the noise in dB associated with each sinusoid(s) if a vector, or for all sinusoids, if a scalar. N is the number of points; f s is assumed to be 1 kHz. The output waveform is in x and t is a time vector useful for plotting.

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Example 3.11 Use fft to find the magnitude spectrum of a waveform consisting of a single 250-Hz sine wave and white noise with an SNR of -14 dB. Calculate the Fourier transform of this waveform and plot the magnitude spectrum. Solution: Use sig_noise to generate the waveform, take the Fourier transform using fft, obtain the magnitude using abs, and plot. N = 1024; % Number of data points fs = 1000; % 1 kHz fs assumed by sig_noise f = (0:N-1)*fs/(N-1); % Frequency vector for plotting % Generate data using sig_noise: f = 250; SNR = -14 dB; N = 1024 [x,t] = sig_noise (250,-14,N); % Generate signal and noise % Xf = fft(x); % Calculate FFT Mf = abs(Xf); % Calculate the magnitude plot(f,Mf); % Plot the magnitude spectrum ……..label and title…….
Example 3.11 Results The spectrum shows a peak related to the 250- Hz sine wave. For reasons described in the next chapter, the spectrum above f s /2 (i.e. 500 Hz) is a mirror image of the lower half of the spectrum. Ideally the background spectrum should be a constant value since the background noise is white noise, but the background produced by noise is highly variable with occasional peaks that could be mistaken for signals. 0 200 400 600 800 1000 0 20 40 60 80 100 120 140 160 Spectrum (symmetric about fs/2) Frequency (Hz) Magnitude

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Summary: Sinusoids The sinusoid (i.e., A cos ( ωt + θ )) is a unique signal with a number of special properties. 1. Sinusoids have energy at only one frequency: the frequency of the sinusoid. This property is unique to sinusoids. 2. A sinusoid can be completely defined by three values: its amplitude A , its phase θ , and its frequency ω (or 2 πf ). 3. Any periodic signal can be broken down into a series of harmonically related sinusoids, although that series might have to be infinite in length. 4. Reversing that logic, any periodic signal can be reconstructed from a series of sinusoids. 5. The sinusoidal representation of a signal can be complete and works in both directions: signals can be decomposed into a (possibly infinite) number of sinusoids and can also be accurately reconstructed from that series
Special Sinusoidal Properties (cont): 6. Since sinusoids have energy at only one frequency, sinusoids can easily be represented in the frequency domain as a magnitude and phase at a given frequency. 7. Sinusoids can serve as intermediaries between the time domain representation of a signal and its frequency domain representation. 8. The calculus operations of differentiation and integration change only the magnitude and phase of a sinusoid. 9. Only the magnitude and phase of a sinusoid can be altered by a linear system. 10. Finally, harmonically related sinusoids are orthogonal so they do not influence one another.

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• Fall '15
• Lipowsky,Herbert
• Fourier Series, Fourier Series Analysis, Fourier Series Equations

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