What would be a reasonable choice of the quantization step Δ?

Transcription

1 CE 108 HOMEWORK 4 EXERCISE 1. Suppose you are samplng the output of a sensor at 10 KHz and quantze t wth a unform quantzer at 10 ts per sample. Assume that the margnal pdf of the sgnal s Gaussan wth mean of 0 Volts and varance of 4 Volts. What s the t rate of the quantzed sgnal? 100 Kts/s What would e a reasonale choce of the quantzaton step Δ? For example, we could choose X m 4 x 8 Volts. Then, Δ X m / Volts What s the power of the quntzaton error? (Assume that the hgh rate hypothess holds). e Δ Volts What s the resultng quantzaton SNR? SNR db Wth your choce of Δ, what s the proalty that a sample s n the overload zone? Ths s P( x >4 x ). For a Gaussan random varale ths can e computed usng the error functon and s equal to EXERCISE. Consder a random varale x wth pdf unform n [-1,1]. Suppose you perform quantzaton wth 3 ts usng a mdrse quantzer. Compute the theoretcal varance of the quantzaton nose, as dvded nto granular and overload zone, choosng X m 0.5, X m 1, and X m. Addtonally, compute the resultng quantzaton SNR. Consder frst the case X m 1. In ths case, the proalty of eng n the overload zone s 0, hence the error s only due to the granular zone. In ths case, 4 Δ 1 e x Δ Δ dx 4 Δ3 ( 1 1 Δ3 3. Snce Δ X m/ 3 1/4, we otan )Δ e Snce the varance of the sgnal s /11/3, we have:

2 SNR10log x db. e f x (x) e(x) For the case X m 0.5 we wll need to also consder the overload zone. Snce Δ X m / 3 1/8, the granular zone gves an error of e The error n the 1536 overload zone can e computed as: e,ol 1 1 x 4Δ Δ 1 dx x 7 9 /16 dx x dx Overall, the 4 Δ 16 1/ 1/ error varance s almost 0.06, whch s much larger (more than 10 tmes) than efore. Ths s due to the overwhelmng overload zone. The SNR s equal to 7.4 db. f x (x) X m 1 e(x) X m 0.5 In the case X m, we have Δ X m / 3 1/. Hence, for x>0, only quantzaton levels are n the area where the varale has non-null proalty. There s no overload error, ut we expect a larger granular error. We need to modfy the equaton for the error varance as follows: e 1 1 Δ ( )Δ x Δ Δ dx Δ3 1 Δ tmes larger than n the case X m 1. Now, SNR1 db , whch s aout 4

3 f x (x) e(x) X m EXERCISE 3. Consder a sgnal wth non-unform margnal dstruton, whch we need to quantze wth 8 ts. Suppose that the optmal quantzaton thresholds are (for x>0) 0.01 (assume that the pdf of the sgnal s symmetrc). Fnd a compoundng functon g(x) such that the compounded sgnal can e quantzed usng a unform quantzer. Ths would e any monotone functon such that (for x>0), g( )g( )(-1)Δ for any choce of Δ. For example, g(x)log ( x /0.01) sgn(x) (g(0)0) would do the trck wth Δ1. EXERCISE 4. (GRADS) Prove that, under the hypothess of hgh rate, the optmal choce of value y for the nterval [-1, ] s the mdpont: (-1 + )/.

4 The optmal value of y s y xf x ( x)dx. In the hgh rate case we assume that f x (x) s ( x)dx 1 f x constant wthn [-1, ]. Let f e such constant value. Then, y 1 1 xf dx f dx 1 1 xdx dx ( ) + Prove that for an optmal quantzer, the quantzaton error has mean equal to 0. Snce y s optmal, t s y E[ e] E[ x y ] E x xf x ( x)dx xf x ( x)dx. Now, wthn [-1, ]: ( x)dx 1 f x 1 xf x ( x)dx xf x ( x)dx f x ( x ) x dx f x ( x)dx f x ( x)dx xf x ( x)dz f x ( x)dx 1 xf x ( x)dx xf x ( x)dx 0 ( x)dx f x EXERCISE 5. (GRADS) 1 x, x 1 Consder a varale x wth the followng trangular pdf: f x ( x). 0, x >1 Fnd a compoundng functon g(x) that transform x nto a unform random varale. Let zg(x). Then, f z (g(x)) f x (x)/ g(x). We want f z (z) to e unform (constant) for all ponts z such that g -1 (z) 1. In other words, wthn ths nterval, t must e g(x) f x (x)/c, where C s a constant. By ntegraton, and forcng g(0)0 and C1, we otan for -1<x<1: g( x) x x sgn(x), whch gves a varale zg(x) unform n [-0.5,0.5]. Suppose you quantze the transformed varale wth a unform quantzer wth 3 ts wth no overload zone. What are the correspondng (non-unform) quantzaton thresholds,x for the orgnal varale? 1

5 We need to fnd the nverse of g(x): for -1<x<1, g 1 ( z) 1 1 z sgn(z). For z>0, the quantzaton thresholds are,z -3, and the correspondng thresholds for x are thus: 1,x 0.134;,x 0.9; 3,x 0.5; 4,x 1. EXERCISE 6. Consder a sgnal x(n), sampled at F100 Hz, and suppose you quantze t usng (1) scalar quantzaton (4 ts per sample) and () vector quantzaton (quantzng vector of samples and assgnng 8 ts per vector). 1. Compute the t rate n the two cases. It s the same (400 ts/s). Prove that the expected quadratc quantzaton error usng vector quantzaton cannot e hgher than n the scalar quantzaton case (assumng that the scalar and the vector quantzer are optmal). It s ecause, gven an optmal scalar quantzer wth nterval set B{ }, you can always construct an dentcal separale vector quantzer, defned y BxB (.e., wth assgnment regons of the type [-1, ]x[ j-1, j ]). Hence, the error of the optmal vector quantzer s at most as large as the error of ths vector quantzer, whch s dentcal to the error of the scalar quantzer. EXERCISE 7. Consder a -D vector quantzer wth y 1 (1,), y (1,4), y 3 (-1,), y 4 (0,-). 1. Show wth a graph the optmal assgnment regons {V }. y y 3 y 1 y 4. Quantze and compute the emprcal quadratc error for the followng sgnal, assumng that you quantze groups of two samples at a tme: x{ } [-4-3] y 4 [0,-] (e 17) [- -1] y 4 [0,-] (e 5) [0 1] y 1 [1,] (e ) (same error s otaned wth y 3 ) [ 3] y 1 [1,] (e ) (same error s otaned wth y )

6 [4 5] y [1,4] (e 10) EXERCISE 8. (GRADS) 1. Prove that at each step of the LBG algorthm to desgn a vector quantzer the expected quadratc norm of the error E[ e ] ( x y ) V f x ( x)dx can never ncrease. (Rememer that the LBG algorthm can e used when the jont pdf f x (x) of the sgnal s known). At each step of the LBG algorthm, we mnmze the expected quadratc error, ether over the set of {V } (keepng the {y } constant) or over the set of {y } (keepng the {V } constant). Ovously, the error can never ncrease. E.g., suppose that at a certan pont we have chosen a certan set {V } and a certan set of {y }, whch gves an expected quadratc error of e. Now we fnd the {y } that mnmze the expected quadratc error for fxed {V }. The error cannot e larger than e otherwse, we may just keep the prevous {y }!. Prove that at each step of the k-mean algorthm to desgn a vector quantzer, the sample mean of the quadratc norm of the error ( x k y ) can never xk V ncrease. (In ths case, we start from a tranng sample {x k }). Same as efore, only that now, at each step of the algorthm, we mnmze ( x k y ), ether over {V } or over {y }. xk V EXERCISE 9. (GRADS - OPTIONAL) We want to desgn a quantzer wth ts for an exponental random varale wth, such that 0 0 and 4. Gven the followng choce of s: [0, 0., 0.6, 0.8, ], fnd the optmal choce of y s y 1 1 x 1 x xe x e d + e x d x (ntegraton y parts) 1 e d e x e 1 e 1 + e e 1 e e e 1 Hence, y [0.0983, , ,.8000]. e 1 e 1 e 1 1 +

7 Gven the set of y s gven y your answer, fnd the optmal set of s y + y +1 (except for 0 and 4 that are fxed). Hence, [0, 0.458, , 1.749]. Now terate, alternatng etween the desgn of the y s and of the s, tll convergence. Ths s the generalzed Lloyd s method for optmal quantzer desgn. Iteratng, I otaned the followng optmal values: : [0, , 3.543, , ] y : [0.660,.3560, , ]. f x (x) x