1. Lecture 2 Outline: Analog-to-Digital and Back Bridging the analog and digital divide
Announcements:
Poll for discussion section and OHs: please respond
First HW posted tonight
Lectures going forward: PPT slides and reader material
PPT slides introduce main take-away ideas from lectures at a high level, which are repeated in more detail in the lecture
Lectures don’t perfectly follow course reader. You are responsible for lecture material and reader sections covered in lecture (will provide pages before exams)
Sampling vs. Analog-to-Digital Conversion (ADC)
Sampling and ADC
Reconstruction and Digital-to-Analog Conversion
Nyquist Sampling Theorem
Quantization

2. Review of Last Lecture Course Overview EE102a Review
Signals and Systems in the Time Domain
Continuous, Discrete, and Hybrid
Continuous-Time Signals in the Frequency Domain
Fourier Series and Fourier Transforms
Discrete Time Signals in the Frequency Domain
Duality Relationships
Connections between Continuous/Discrete Time
Filtering and Convolution
Periodic Signals
Energy, Power, and Parseval

3. Sampling and Reconstruction vs
Sampling and Reconstruction vs. Analog-to-Digital and Digital-to-Analog Conversion
Sampling: converts a continuous-time signal to a continuous-time sampled signal
Reconstruction: converts a sampled signal to a continuous-time signal.
Analog-to-digital conversion: converts a continuous-time signal to a discrete-time quantized or unquantized signal
Digital-to-analog conversion. Converts a discrete-time quantized or unquantized signal to a continuous-time signal. Ts
2Ts
3Ts
4Ts
-3Ts
-2Ts
-Ts Ts
2Ts
3Ts
4Ts
-3Ts
-2Ts
-Ts 1 2 3 4 -3 -2 -1
Each level can
be represented
by 0s and 1s 1 2 3 4 -3 -2 -1

4. Applications Capture: audio, images, video
Storage: CD, DVD, Blu-Ray, MP3, JPEG, MPEG
Signal processing: compression, enhancement and synthesis of audio, images, video
Communication: optical fiber, cell phones, wireless local-area networks (WiFi), Bluetooth
Applications: VoIP, streaming music and video, control systems, Fitbit, Occulus Rift

5. Sampling Sampling (Time): =
Sampling (Frequency): x(t)p(t) X(jw)*P(jw)/(2p)
Analog-to-Digital Conversation (ADC)
Setting xd[n]=xs(nTs) yields Xs(ejW) with W=wTs
x(t)
nd(t-nTs) =
xs(t) Ts
2Ts
3Ts
4Ts
-3Ts
-2Ts
-Ts
-3Ts
-2Ts
-Ts Ts
2Ts
3Ts
4Ts
X(jw) * =
nd(w-(2pn/Ts))
Xs(jw)
-2p Ts 2p Ts
-2p Ts 2p Ts

6. Reconstruction Frequency Domain: low-pass filter
Time Domain: sinc interpolation
Digital-to-Analog Conversation (DAC)
LPF applied to Xs(ejW) and then converted to continuous time (w=W/Ts) recovers sampled signal
H(jw)
Xs(jw)
-2p Ts
Xs(jw) 2p
Xr(jw)
H(jw) 1 -W W w w
-2p Ts 2p Ts

7. Nyquist Sampling Theorem
A bandlimited signal [-W,W] radians is completely described by samples every Tsp/W secs.
The minimum sampling rate for perfect reconstruction, called the Nyquist rate, is W/p samples/second
If a bandlimited signal is sampled below its Nyquist rate, distortion (aliasing occurs)
X(jw)
X(jw)
Xs(jw) -W W
-2W W W
2W=2p/Ts

8. Quantization x(t) xQ(nTs) A … -A+kD … -A+2D -A+D -ATs
2Ts
Divide amplitude range [-A,A] into 2N levels, {-A+kD}, k=0,…2N-1
Map x(t) amplitude at each Ts to closest level, yields xQ(nTs)=xQ[n]
Convert k to its binary representation (N bits); converts xQ[n] to bits

9. Main Points
Sampling bridges the analog and digital worlds, with widespread applications in the capture, storage, and processing of signals
Sampling converts continuous-time signals to sampled signals
Multiplication with delta train in time, convolution with delta train in frequency
ADC converts a continuous-time signal to a discrete-time signal or bits
Reconstruction recreates a continuous-time signal from its samples
Multiplication with LPF in frequency, sinc interpolation in time
DAC recreates a continuous-time signal from a discrete-time signal or bits
Reconstruction in the frequency domain entails low-pass filtering; in the time-domain it entails convolution with a sinc function.
A bandlimited signal of bandwidth W sampled at or above its Nyquist rate of 2W can be perfectly reconstructed from its samples
Quantization converts a discrete-time signal to bits by mapping its values to a finite number of levels, which introduces noise