CHAPTER 5 Digital Processing of Continuous- Time Signal Wangweilian School of Information Science and Technology Yunnan University

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 2 Outline Sampling of Continuous-Time Signals Sampling of Bandpass Signals Analog Lowpass Filter Design Summary

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 3 Sampling of Continuous-Time Signals Effect of Sampling in the Frequency-Domain –In the time-domain: T — sampling period — sampling frequency –In the frequency-domain: continuous-time: discrete-time:

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 4 Sampling of Continuous-Time Signals Mathematical analysis: –Step 1: a multiplication of the continuous-time signal by a periodic impulse train :

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 5 Sampling of Continuous-Time Signals –Using Fourier series: — the angular sampling frequency The multiplication in the time-domain — the period function in the frequency-domain

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 6 Sampling of Continuous-Time Signals The sampling theorem –The Nyquist condition: –The Nyquist frequency: – is a ideal lowpass filter with a gain T and a cutoff frequency : –oversampling — undersampling — critical sampling

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 7 Sampling of Continuous-Time Signals –Step 2 : the relation between the DTFT of and the FT of Since ， That is normalizing the time axis — scaling the frequency axis

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 8 Sampling of Continuous-Time Signals Recovery of the Analog Signal –the ideal lowpass filter : –Assuming, the reconstructed continuous-time signal : –Ideal interpolation — ideal lowpass filter

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 9 Sampling of Bandpass Signals Bandpass signal : The sampling frequency : Where, and M is any integer. The FT of the impulse-sampled signal :

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 10 Analog Lowpass Filter Design Filter Specifications –The magnitude: Where is the passband / stopband edge frequency –Ripples — : –The loss function or attenuation function:

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 11 Analog Lowpass Filter Design Filter Specifications in the normalized form –The maximum value of the magnitude in passband is 1 –The passband ripple is –The maximum passband gain or the minimum passband loss is 0 dB –The maximum stopband ripple is –The minimum stopband attenuation is Two additional parameters –The transition ratio or selectivity parameter: –The discrimination parameter:

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 12 Analog Lowpass Filter Design Butterworth Approximation –The magnitude-squared response ( Nth order ): –The gain of the Butterworth filter: –The order N is:

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 13 Analog Lowpass Filter Design Butterworth Approximation –The transfer function of the Butterworth lowpass filter: –The poles: –The order N of the transfer function must be a integer.

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 14 Analog Lowpass Filter Design Chebyshev Approximation –The approximation error: The difference between the ideal brickwall characteristic and the actual response Minimized over a prescribed band of frequencies –The magnitude error is equiripple in the band: passbandstopband Type 1 the magnitude characteristic equiripplemonotonic Type 2 the magnitude response monotonicequiripple

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 15 Analog Lowpass Filter Design Type 1 Chebyshev Approximation –The magnitude-squared response: –The Chebyshev polynomial: –A recurrence relation:

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 16 Analog Lowpass Filter Design Type 1 Chebyshev Approximation –The transfer function: –The order N: –The poles:

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 17 Analog Lowpass Filter Design Type 2 Chebyshev Approximation –The magnitude-squared response: –The transfer function: –The zeros on the :

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 18 Analog Lowpass Filter Design Type 2 Chebyshev Approximation –If N is odd, then for the zeros is at. –The poles:

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 19 Analog Lowpass Filter Design Elliptic Approximation –The square-magnitude response: is a rational function of order N satisfying the property. –The order:

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 20 Analog Lowpass Filter Design Linear-Phase Approximation –The all-pole transfer function: –The Bessel polynomial: –The coefficients of the Bessel polynomial:

云南大学滇池学院课程：数字信号处理 Digital Processing of Continuous-Time signal 21 Summary A discrete-time signal is obtained by uniformly sampling a continuous-time signal. The discrete-time representation is unique if the sampling frequency is greater than twice the highest frequency contained in the continuous-time signal, and the latter can be fully recovered from its discrete-time equivalent by passing it through an ideal analog lowpass reconstruction filter with a cutoff frequency that is half the sampling frequency. The specifications of the analog filters are usually given in terms of the locations of the passband and stopband edge frequencies.