Fourier and Laplace transforms, convolution, sampling, transfer functions, block diagrams, stability, Bode and root locus.
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- What is a Fourier series?
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A representation of a periodic signal as an infinite sum of sinusoidal components (sines and cosines) at harmonic frequencies.- Define the Fourier transform.
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A mathematical tool that decomposes a signal into its frequency components, transforming it from the time domain to the frequency domain.- State the linearity property of the Fourier transform.
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If x(t) has transform X(f) and y(t) has transform Y(f), then a*x(t) + b*y(t) has transform a*X(f) + b*Y(f).- What is the time-shift property of the Fourier transform?
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If x(t) has transform X(f), then x(t - t0) has transform exp(-j*2*pi*f*t0)*X(f).- What is the frequency-shift property of the Fourier transform?
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If x(t) has transform X(f), then exp(j*2*pi*f0*t)*x(t) has transform X(f - f0).- State the scaling property of the Fourier transform.
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If x(t) has transform X(f), then x(a*t) has transform (1/|a|)*X(f/a).- What does Parseval's theorem state for the Fourier transform?
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Energy of a signal in time domain equals energy in frequency domain: integral of |x(t)|^2 dt = integral of |X(f)|^2 df.- Define the Nyquist frequency.
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The highest frequency that can be represented in a sampled signal, equal to half the sampling frequency (fs/2).- What is aliasing in signal sampling?
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A phenomenon where frequency components above the Nyquist frequency appear as lower-frequency components in the sampled signal.- Define the Laplace transform.
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An integral transform: X(s) = integral from 0 to infinity of x(t)*exp(-s*t) dt, where s is a complex variable.- What is the Laplace transform of the unit step u(t)?
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1/s, valid for Re(s) > 0.- What is the Laplace transform of exp(-a*t)*u(t)?
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1/(s + a), valid for Re(s) > -a.- What is the Laplace transform of sin(omega*t)*u(t)?
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omega/(s^2 + omega^2), valid for Re(s) > 0.- What is the Laplace transform of cos(omega*t)*u(t)?
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s/(s^2 + omega^2), valid for Re(s) > 0.- State the linearity property of the Laplace transform.
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If L{x(t)} = X(s) and L{y(t)} = Y(s), then L{a*x(t) + b*y(t)} = a*X(s) + b*Y(s).
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