EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 15-Sept, 1998EE421, Lecture 031 Lecture 3: Quantization l The last major stage of an A/D converter is the conversion of the sampled signal to a digital signal: sampled analog signal quantized signal B bits/sample A/D converter v in v out example mapping from input to output

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 15-Sept, 1998EE421, Lecture 032 Quantization l Key parameters for an A/D converter –full-scale voltage range: R –number of bits: B –number of quantization levels: 2 B –quantization width: Q = R / 2 B bit unsigned binary code 0 Q 2Q 3Q 4Q 5Q 6Q 7Q 8Q R t input signal quantized signal

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 15-Sept, 1998EE421, Lecture 033 Quantization Error l The rounding operation replaces each sample by the nearest quantized value. –quantization error e then lies in the interval: –maximum error (magnitude): Q/2 v out error V in + Q/2V out - Q/2 Q/2 -Q/2 average error = 0 average square error = Q 2 /12

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 15-Sept, 1998EE421, Lecture 034 Quantization Error l Signal to Noise Ratio (SNR) or Dynamic Range: –maximum signal range: R –maximum noise range: Q –6 dB per bit Example: The human ear has a dynamic range of about 100 dB. This requires approximately 16 bits which is the number of bits associated with “CD quality” recording.

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 15-Sept, 1998EE421, Lecture 035 Quantization Error l Large dynamic ranges can be obtained with fewer bits, provided: –the sampling rate is increased; and –noise-shaping is performed. Advanced topic: noise shaping quantizers

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 15-Sept, 1998EE421, Lecture 036 D/A Conversion l unipolar natural binary: l bipolar offset binary: l two’s complement: DAC b1b1 b2b2 b3b3 bBbB input bits x analog output R

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 15-Sept, 1998EE421, Lecture 037 D/A Conversion l Horner’s algorithm: sum = 0; b[1] = 1-b[1]; for i=B, B-1, B-2, …, 1 sum = 0.5*(sum + b[i]); end; x = R*(sum - 0.5);

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 15-Sept, 1998EE421, Lecture 038 A/D Conversion l Successive approximation (rounding) + - b1b1 b1b1 b2b2 b2b2 b3b3 b3b3 bBbB bBbB successive approximation register DAC ADCcomparator analog input digital output R xQxQ x + Q/2 u(x-x Q )

EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 15-Sept, 1998EE421, Lecture 039 A/D Conversion l Example: –4 bits –bipolar two’s complement –R = 5 V (Q = V) –input voltage : -1.8V