Analog-to-Digital and Back

Transcription

1 Lecture 2 Outine: Anaog-to-Digita and Back Bridging the anaog and digita divide Announcements: Po for discussion section and OHs: pease respond First HW posted tonight Lectures going forward: PPides and reader materia PPides introduce main take-away ideas from ectures at a high eve, which are repeated in more detai in the ecture Lectures don t perfecty foow course reader. You are responsibe for ecture materia and reader sections covered in ecture (wi provide pages before exams) Samping vs. Anaog-to-Digita Conversion (ADC) Samping and ADC Reconstruction and Digita-to-Anaog Conversion Nyquist Samping Theorem Quantization

2 Review of Last Lecture Course Overview EE102a Review Signas and Systems in the Time Domain Continuous, Discrete, and Hybrid Continuous-Time Signas in the Frequency Domain Fourier Series and Fourier Transforms Discrete Time Signas in the Frequency Domain Fourier Series and Fourier Transforms Duaity Reationships Connections between Continuous/Discrete Time Fitering and Convoution Periodic Signas Energy, Power, and Parseva

3 Samping and Reconstruction vs. Anaog-to- Digita and Digita-to-Anaog Conversion Samping: converts a continuous-time signa to a continuoustime samped signa Reconstruction: converts a samped signa to a continuoustime signa Anaog-to-digita conversion: converts a continuous-time signa to a discrete-time quantized or unquantized signa Each eve can be represented by 0s and 1s Digita-to-anaog conversion. Converts a discrete-time quantized or unquantized signa to a continuous-time signa

4 Appications Capture: audio, images, video Storage: CD, DVD, Bu-Ray, MP3, JPEG, MPEG Signa processing: compression, enhancement and synthesis of audio, images, video Communication: optica fiber, ce phones, wireess oca-area networks (WiFi), Buetooth Appications: VoIP, streaming music and video, contro systems, Fitbit, Occuus Rift

5 Samping Samping (Time): x(t) å n d(t-n ) = x s (t) 0 Samping (Frequency): x(t)p(t) * X(jw) 1 å n d(w-(2pn/ )) = X(jw)*P(jw)/(2p) X s (jw) -2p 2p -2p Anaog-to-Digita Conversation (ADC) 2p Setting x d [n]=x s (nts) yieds X s (e jw ) with W=w

6 Reconstruction Frequency Domain: ow-pass fiter X s (jw) H(jw) X r (jw) 1 H(jw) X s (jw) -2p 0 2p -W W w -2p 0 2p w Time Domain: sinc interpoation x r ( t) = å xs( nts ) h( t - nts ) = å n=- n=- x ( nt ) s s æ t - nt sinc ç è Ts s ö ø Digita-to-Anaog Conversation (DAC) LPF appied to X s (e jw ) and then converted to continuous time (w=w/ ) recovers samped signa

7 Nyquist Samping Theorem A bandimited signa [-W,W] radians is competey described by sampes every p/w secs. The minimum samping rate for perfect reconstruction, caed the Nyquist rate, is W/p sampes/second If a bandimited signa is samped beow its Nyquist rate, distortion (aiasing occurs) X(jw) X(jw) X s (jw) -W 0 W -2W W 0 W 2W=2p/

8 Quantization A -A+kD x Q (n ) x(t) -A+2D -A+D -A 0 2 Divide ampitude range [-A,A] into 2 N eves, {-A+kD}, k=0, 2 N -1 Map x(t) ampitude at each to cosest eve, yieds x Q (n )=x Q [n] Convert k to its binary representation (N bits); converts x Q [n] to bits

9 Main Points Samping bridges the anaog and digita words, with widespread appications in the capture, storage, and processing of signas Samping converts continuous-time signas to samped signas Mutipication with deta train in time, convoution with deta train in frequency ADC converts a continuous-time signa to a discrete-time signa or bits Reconstruction recreates a continuous-time signa from its sampes Mutipication with LPF in frequency, sinc interpoation in time DAC recreates a continuous-time signa from a discrete-time signa or bits Reconstruction in the frequency domain entais ow-pass fitering; in the time-domain it entais convoution with a sinc function. A bandimited signa of bandwidth W samped at or above its Nyquist rate of 2W can be perfecty reconstructed from its sampes Quantization converts a discrete-time signa to bits by mapping its vaues to a finite number of eves, which introduces noise