ece306-3 - ECE 306 Sampling of Analog Signals Quantization...

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1 °±° ²³´µ² ¶ ECE 306 Sampling of Analog Signals Quantization of Continuous-Amplitude Signals °± ²³´µ¶·´¸´¹º³» ¼ ³½¸¾¿ ´¸¶³ ¶ÀÁ¹ÃÄ»¾½¿¼Àº´À½½¿ ´Àº ŽĶ¿¾Ã½À¾ ¶³ ƹ³µ ƹùÀ¶ °±° ²³´ µ² °±° ²³´µ² · Sampling of Analog Signals Example: a. Find the minimum sampling rate required to avoid aliasing. b. If , What is the discrete-time signal after sampling? c. If , What is the discrete-time signal after sampling? d. What is the frequency F of a sinusoidal that yields sampling identical to obtained in part c? a The minimum sampling rate is ( ) 3cos100 a x t t π = Solution: 100 π Ω = 50 Hz F = 2 100Hz s F F = = and the discrete-time signal is ( ) ( ) 3cos 3cos 3cos2 100 1 100 2 a x n x nT n n n π π π = = = = 200 s F Hz = 75 s F Hz =
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2 °±° ²³´µ² ² Sampling of Analog Signals b and in part in (c). Hence Solution: If , the discrete-time signal is 200 s F Hz = ( ) 3cos 3cos 3cos2 100 1 200 2 4 x n n n n π π π = = = c If , the discrete-time signal is 75 Hz s F = 2 100 4 1 ( ) 3cos 3cos 3cos 2 3cos2 3 3 75 3 x n n n n n π π π π π = = = - ° ± = ² ³ ´ µ d For the sampling rate , 75 Hz s F = 75 s F fF f = = 1 3 f = 75 25Hz 3 F = = ( ) 3cos2 3cos50 a y t Ft t π π = = So, the analog sinusoidal signal is °±° ²³´µ² ¸ The Sampling Theorem We must have some information about the analog signal especially the frequency content of the signal, to select the sampling period T or sampling rate F s . For example A speech signal goes below around 20Khz. A TV signal is up to 5Mhz. Any analog signal can be represented as sum of sinusoids of different amplitudes, frequencies, and phases. 1 ( ) cos(2 ) N a i i i i x t A Ft π θ = = + where N the number of frequency components. Suppose that Nth frequency do not exceed the largest frequency max F max i F F <
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3 °±° ²³´µ² ¹ The Sampling Theorem To avoid the aliasing problem, is selected so that The analog signal should be in the range of or in radians The sampling rate is called the Nyquist rate .
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