5.5.4
Real world continuous hai (voltage smoothly time aur amplitude mein vary karta hai). CPU sirf discrete numbers samajhta hai. Ek ADC (Analog-to-Digital Converter) continuous voltage ko ek number mein badalta hai; DAC yeh kaam ulta karta hai.
Do sawaal quality decide karte hain:
Hum amplitude ko kitne fine tarike se slice karte hain? → resolution (bits).Hum time mein kitni baar measure karte hain? → sampling rate (Hz).
Agar koi bhi galat ho toh digital copy ek jhooth hai. Nyquist woh rule hai jo batata hai ki frequency ke baare mein jhooth na bolne ke liye minimum speed kya honi chahiye.
N 2 N Resolution sabse chhota voltage step hai jo woh represent kar sakta hai, ise LSB (Least Significant Bit) kehte hain.
YEH FORMULA KYUN? Har bit ek yes/no switch hai. N 0 2 N − 1 V r e f 2 N 2 N − 1
Worked example 12-bit ADC,V r e f = 3.3
2 12 = 4096 LSB = 3.3/4095 ≈ 0.806 mV Yeh step kyun? ~0.8 mV se chhota koi bhi input change dikh nahi sakta — woh same code mein hi land karta hai.1.65 V ke liye code: code = round ( 3.3 1.65 × 4095 ) = 2048 Round kyun? ADC nearest level pe snap karta hai (mid-tread quantizer).
Hum har T s Sampling rate f s = 1/ T s
Minimum speed kyun zaroori hai
Ek sine wave ko apna upar-neeche dikhane ke liye kam se kam do samples per period chahiye. Kam samples ke saath, ek fast wave ek slow wave se alag nahi dikh sakti — woh khud ko disguise kar leti hai. Yeh disguise aliasing hai.
Worked example Aliasing in actionSignal f = 1800 f s = 1000
Nyquist = 500 1800 > 500 Kyun? Yeh f s > 2 f ma x
round ( 1800/1000 ) = 2 f a l ia s = ∣1800 − 2000∣ = 200 Result: ek 1.8 kHz tone ek fake 200 Hz tone ke roop mein record hota hai. Fix: ADC se pehle ek anti-aliasing low-pass filter jo 500 Hz se upar sab kuch khatam kar de.
Worked example Audio CD designHuman hearing ~20 kHz tak hai → f s > 40 44.1 kHz use karta hai. Extra margin kyun? Real anti-alias filters perfectly sharp nahi hote; yeh gap (40→44.1 kHz) filter ko roll off karne ki jagah deta hai.
Codes leta hai aur voltage output karta hai V o u t = code × LSB reconstruction (smoothing) low-pass filter staircase steps ko hataata hai taaki ek continuous wave recover ho sake.
Intuition Reconstruction ko bhi filter kyun chahiyeDAC har sample ko T s f s /2
Common mistake "Nyquist kehta hai exactly2 f ma x Yeh sahi kyun lagta hai: formula mein literally 2 hai. Trap: exactly 2 f strict hai: f s > 2 f ma x ≥ 2.5 ×
Common mistake "Zyada bits aliasing fix karte hain."Yeh sahi kyun lagta hai: zyada bits = higher fidelity, toh zaroor zyada = har jagah better. Fix: bits = amplitude precision; aliasing ek time/frequency problem hai. Koi bhi bits ek aisi frequency recover nahi kar sakti jo aapne under-sample ki ho. Aapko faster f s
= V r e f / 2 N Yeh sahi kyun lagta hai: 2 N Fix: 2 N 2 N − 1 gaps hote hain, toh exact LSB = V r e f / ( 2 N − 1 ) / 2 N
Recall Feynman: ek 12-saal ke bacche ko explain karoSocho ek moving wave ko draw karna by apni pencil paper par thodi thodi der mein dot karke.
Resolution = tumhare paper par kitni height-lines hain (zyada lines = smoother heights). Sampling rate = tum kitni baar dot karte ho. Agar koi wave fast upar-neeche hilti hai lekin tum bahut slow dot karo, toh tumhare dots ek alag, slow wave banate hain — yeh ek magic trick hai jise aliasing kehte hain jahan fast wave ek slow wave ki tarah chhupta hai. Rule: kam se kam har wiggle mein do baar dot karo. DAC woh dost hai jo tumhare dots ko wapas ek smooth wave mein jodta hai.
"Bits = Height, Rate = Time, Two-times = No Crime."
Height → resolution/LSB. Time → sampling. Two-times → f s > 2 f ma x
ADC resolution (bits) kya determine karta hai? Amplitude precision — levels ki sankhya 2 N = V r e f / ( 2 N − 1 )
N-bit ADC ke LSB ka formula? V r e f / ( 2 N − 1 ) ≈ V r e f / 2 N
3.3 V ref ke saath 12-bit ADC ka LSB? ≈ 0.806 3.3/4095
Nyquist criterion batao. f s > 2 f ma x
Nyquist frequency kya hai? f s /2 f s
Aliasing kya hai? f s /2
1000 Hz par sample kiya 1800 Hz ka alias frequency? ∣1800 − 2 ⋅ 1000∣ = 200
Aliasing kaise rokein? ADC se pehle anti-aliasing low-pass filter (cut-off below f s /2
CD 40 kHz ki jagah 44.1 kHz kyun use karta hai? Anti-alias filter ke roll-off ke liye 2 × 20
N-bit ADC ka ideal SNR? 6.02 N + 1.76
Quantization noise ke liye q 2 /12 Width q q 2 /12
DAC ka reconstruction filter kya karta hai? Zero-order-hold staircase ko smooth karta hai, f s /2
Kya bits add karne se aliasing fix hoti hai? Nahi — bits amplitude precision fix karte hain; aliasing ke liye faster f s
Quantization Noise & SNR Anti-Aliasing Filters Zero-Order Hold & Reconstruction Fourier Transform & Frequency Domain Fixed-Point vs Floating-Point in Embedded DSP Successive Approximation vs Sigma-Delta ADC Sensor Interfacing on Microcontrollers
Continuous real-world signal
Nyquist fs greater than 2 fmax
