Q5Signals and Systems

Question

Find the Nyquest rate and discrete signal x(n) for the following continuous time signals: (a) $x (t) = sin (4 \times 1 0^{3} π t) / π t$ (b) $x (t) = 1 + cos (2000 π t) + sin (400 π t)$ (c) $x (t) = sin c (50 π t) + sin c (100 π t)$

Answer 1

The Nyquist rate is defined as $2 f_{ma x}$ , where $f_{ma x}$ is the highest frequency component present in the signal.

(a) $x (t) = sin (4 \times 1 0^{3} π t) / π t$ This is a sinc function, representing an ideal low-pass filter in the frequency domain with cutoff frequency $ω_{m} = 4000 π$ rad/s. $2 π f_{m} = 4000 π ⟹ f_{m} = 2000$ Hz. Nyquist rate $= 2 f_{m} = 2 \times 2000 = 4000$ Hz.

(b) $x (t) = 1 + cos (2000 π t) + sin (400 π t)$ The signal has three frequency components: $0$ Hz (DC), $1000$ Hz (from $2000 π$ ), and $200$ Hz (from $400 π$ ). The maximum frequency $f_{ma x} = 1000$ Hz. Nyquist rate $= 2 f_{ma x} = 2000$ Hz.

(c) $x (t) = sinc (50 π t) + sinc (100 π t)$ Assuming standard unnormalized definition $sinc (x) = sin (x) / x$ . The highest frequency component comes from the second term, $ω_{m} = 100 π$ rad/s. $2 π f_{m} = 100 π ⟹ f_{ma x} = 50$ Hz. Nyquist rate $= 2 f_{ma x} = 100$ Hz.

Q5Signals and Systems

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