DSP-Lec 01-Sampling and Reconstruction

Chapter p Sampling and Reconstruction Ha Hoang Kha, Ph.D.Click to edit Master subtitle style Ho Chi Minh City University of Technology @ Email: hhkha@hcmut.edu.vn Content ™ Sampling ‰ Sampling theorem ‰ Spectrum of sampling signals ™ Antialiasing prefilter ‰ Ideal Id l prefilter fil ‰ Practical prefilter ™ Analog reconstruction ‰ Ideal reconstructor ‰ Practical reconstructor Ha H Kha Sampling and Reconstruction Review of useful equations ™ Linear system Linear system h(t) H(f) x(t ) X(f ) y (t ) = x(t ) ∗ h(t ) Y ( f ) = X ( f )H ( f ) ™ Especially, Especially x(t ) = A cos(2π f 0t + θ ) y (t ) = A | H ( f ) | cos(2π f 0t + θ + arg( H ( f ))) cos(2π f 0t ) ←⎯→ [δ ( f + f ) + δ ( f − f )] sin(2π f 0t ) ←⎯→ ← FT → j[δ ( f + f ) − δ ( f − f )] ™ Trigonometric formulas: cos(a) cos(b) = [cos(a + b) + cos(a − b)] sin( a) sin(b) = − [cos( a + b) − cos( a − b)] sin( a) cos(b) = [sin( a + b) + sin( a − b)] ™ Fourier transform: Ha H Kha FT Sampling and Reconstruction Introduction ™ A typical signal processing system includes stages: ™ The Th analog l signal i l is i digitalized di i li d b by an A/D converter ™ The digitalized samples are processed by a digital signal processor ‰ The digital processor can be programmed to perform signal processing operations such as filtering, spectrum estimation Digital signal processor can be a general purpose computer, DSP chip or other digital hardware ™ The resulting output samples are converted back into analog by a D/A converter Ha H Kha Sampling and Reconstruction Analog to digital conversion ™ Analog to digital (A/D) conversion is a three-step process x(t) Sampler t=nT x(t) x(nT)≡x(n) ( ) ( ) Quantizer xQ(n) Coder A/D converter x(n) t Ha H Kha 11010 n 111 xQ(n) 110 101 100 011 010 001 000 n Sampling and Reconstruction Sampling ™ Sampling is to convert a continuous time signal into a discrete time signal The analog signal is periodically measured at every T seconds signal ™ x(n)≡x(nT)=x(t=nT), ( ) ( ) ( ), n=….-2,, -1,, 0,, 1,, 2,, 3…… ™ T: sampling interval or sampling period (second); ™ fs=1/T: sampling rate or sampling frequency (samples/second or Hz) Ha H Kha Sampling and Reconstruction Sampling-example ™ The analog signal x(t)=2cos(2πt) with t(s) is sampled at the rate fs=4 Hz Find the discrete-time signal g x(n) ( )? Solution: ™ x(n)≡x(nT)=x(n/fs)=2cos(2πn/fs)=2cos(2πn/4)=2cos(πn/2) n x(n) ‐2 ™ Plot the signal Ha H Kha Sampling and Reconstruction Sampling-example ™ Consider the two analog sinusoidal signals x1 (t ) = cos(2π t ), ) x2 (t ) = 2cos(2π t ); t (s) 8 These signals are sampled at the sampling frequency fs=1 Hz Fi d the Find h discrete-time di i signals i l ? Solution: 71 ) = cos(2π n) = cos( π n) fs 81 π = cos((2 − )π n) = cos( n) 4 11 x2 (n) ≡ x2 (nT ) = x2 (n ) = cos(2π n) = cos( π n) fs 81 x1 (n) ≡ x1 (nT ) = x1 (n ™ Observation: x1(n)=x2(n) Ỉ based on the discrete-time signals, we cannot tell which of two signals are sampled ? These signals are called “alias” Ha H Kha Sampling and Reconstruction Sampling-example f2=1/8 Hz f1=7/8 Hz fs=1 Hz Fig: Illustration of aliasing Ha H Kha Sampling and Reconstruction Sampling-Aliasing of Sinusoids ™ In general, the sampling of a continuous-time sinusoidal signal p g rate fs=1/T / results in a discretex(t ) = A cos(2π f 0t + θ ) at a sampling time signal x(n) ™ The sinusoids xk (t ) = A cos(2π f k t + θ ) is sampled at fs , resulting in a discrete time signal xk(n) ™ If fk=f0+kfs, k=0, k=0 ±1, ±1 ±2, ±2 …., then x(n)=xk(n) Proof: ((in class)) ™ Remarks: We can that the frequencies fk=f0+kfs are indistinguishable from the frequency f0 after sampling and hence they are aliases of f0 Ha H Kha 10 Sampling and Reconstruction Analog reconstruction-Example ™ The analog signal x(t)=cos(20πt) is sampled at the sampling frequency fs=40 Hz a) Plot the spectrum of signal x(t) ? b) Find Fi d the h discrete di time i signal i l x(n) ( )? c) Plot the spectrum of signal x(n) ? d) The signal x(n) is an input of the ideal reconstructor, reconstructor find the reconstructed signal xa(t) ? Ha H Kha 17 Sampling and Reconstruction Analog reconstruction-Example ™ The analog signal x(t)=cos(100πt) is sampled at the sampling frequency fs=40 Hz a) Plot the spectrum of signal x(t) ? b) Find Fi d the h discrete di time i signal i l x(n) ( )? c) Plot the spectrum of signal x(n) ? d) The signal x(n) is an input of the ideal reconstructor, reconstructor find the reconstructed signal xa(t) ? Ha H Kha 18 Sampling and Reconstruction Analog reconstruction ™ Remarks: xa(t) contains only the frequency components that lie in the Nyquist interval (NI) [-f [ fs//2, //2 fs/2] /2] sampling p g at fs ideal reconstructor ™ x(t), f0 ∈ NI > x(n) > xa(t), fa=ff0 sampling at fs ideal reconstructor ™ xk(t), fk=f0+kfs > x(n) > xa(t), fa=f0 ™ The Th frequency f fa off reconstructedd signal i l xa(t) ( ) is i obtained b i d by b adding ddi to or substracting from f0 (fk) enough multiples of fs until it lies within the Nyquist interval [[-ffs//2, //2 fs/2] /2] That is f a = f mod( f s ) Ha H Kha 19 Sampling and Reconstruction Analog reconstruction-Example ™ The analog signal x(t)=10sin(4πt)+6sin(16πt) is sampled at the rate 20 Hz Findd the h reconstructed d signall xa(t) ? Ha H Kha 20 Sampling and Reconstruction Analog reconstruction-Example ™ Let x(t) be the sum of sinusoidal signals x(t)=4+3cos(πt)+2cos(2πt)+cos(3πt) x(t) 4+3cos(πt)+2cos(2πt)+cos(3πt) where t is in milliseconds milliseconds a) Determine the minimum sampling rate that will not cause any aliasing effects ? b) To observe aliasing effects, suppose this signal is sampled at half its Nyquist rate Determine the signal xa(t) that would be aliased with x(t) ? Plot the spectrum of signal x(n) for this sampling rate? Ha H Kha 21 Sampling and Reconstruction Ideal antialiasing prefilter ™ The signals in practice may not bandlimitted, thus they must be f l d by filtered b a lowpass l fl filter Fi Ideal Fig: Id l antialiasing ti li i prefilter p filt Ha H Kha 22 Sampling and Reconstruction Practical antialiasing prefilter ™ A lowpass filter: [-fpass, fpass] is the frequency range of interest for the application l (ffmax=ffpass) ™ The Nyquist frequency fs/2 is in the middle of transition region ™ The stopband frequency fstop and the minimum stopband attenuation Astop dB must be chosen appropriately to minimize the aliasing effects effects f s = f pass + f stop Fig: Practical antialiasing lowpass prefilter Ha H Kha 23 Sampling and Reconstruction Practical antialiasing prefilter ™ The attenuation of the filter in decibels is defined as A( f ) = −20 log10 H( f ) (dB) H ( f0 ) where f0 is a convenient reference frequency, typically taken to be at p filter DC for a lowpass ™ α10 =A(10f)-A(f) (dB/decade): the increase in attenuation when f is g byy a factor of ten changed ™ α2 =A(2f)-A(f) (dB/octave): the increase in attenuation when f is changed by a factor of two ™ Analog filter with order N, |H(f)|~1/fN for large f, thus α10 =20N (dB/decade) and α10 =6N (dB/octave) Ha H Kha 24 Sampling and Reconstruction Antialiasing prefilter-Example ™ A sound wave has the form x(t ) = A cos(10π t ) + B cos(30π t ) + 2C cos(50π t ) + D cos(60π t ) + E cos(90π t ) + F cos(125π t ) where t is in milliseconds What is the frequency content of this g ? Which parts p of it are audible and whyy ? signal This signal is prefilter by an anlog prefilter H(f) Then, the output y(t) prefilter is sampled p at a rate of 40KHz and immediatelyy of the p reconstructed by an ideal analog reconstructor, resulting into the final analog output ya(t), as shown below: Ha H Kha 25 Sampling and Reconstruction ... are called “alias” Ha H Kha Sampling and Reconstruction Sampling-example f2=1/8 Hz f1=7/8 Hz fs=1 Hz Fig: Illustration of aliasing Ha H Kha Sampling and Reconstruction Sampling-Aliasing of Sinusoids... and Reconstruction Analog reconstruction- Example ™ The analog signal x(t)=10sin(4πt)+6sin(16πt) is sampled at the rate 20 Hz Findd the h reconstructed d signall xa(t) ? Ha H Kha 20 Sampling and. .. fk=f0+kfs are indistinguishable from the frequency f0 after sampling and hence they are aliases of f0 Ha H Kha 10 Sampling and Reconstruction Sampling Theorem-Sinusoids ™ Consider the analog signal