DSP IC, Solutions. The pseudo-power entering into the adaptor is: 2 b 2 2 ) (a 2. Simple, but long and tedious simplification, yields p = 0.

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1 5 FINITE WORD LENGTH EFFECTS 5.4 For a two-ort adator we have: b a + α(a a ) b a + α(a a ) α R R R + R The seudo-ower entering into the adator is: R (a b ) + R (a b ) Simle, but long and tedious simlification, yields a) In safe scaling, we shall have S. S h(n) a 0 + a + a + a + a 0 n The new scaled values are: a 0 /S, a /S, and a /S. b) In L -norm scaling, we shall have S, where S h(n) a 0 + a + a + a + a 0 n Let c S. The new values are: a 0 /c, a /c, and a /c. 5.7 Rank the different scaling criteria in terms of severity. Most severe is: Safe scaling, then L, and finally, L. 5.8 a) F(e jωt ) 0 0 ( b) We have: F(e jωt ) dωt 0 F(e ) jωt ) dωt F(e jωt ) dωt F(e jωt ) F(e jωt ) G(e jωt ) F(e jωt ) G(e jωt ) dωt

2 F(e jωt ) max G(e jωt ) dωt max F(e jωt ) G(e jωt ) dωt F(e jωt ) G(e jωt ), > Hence, F(e jωt ) G(e jωt ) F(e jωt ) G(e jωt ), > 5.9 The inut x(n) is assumed to be scaled. Hence, the inuts to the multiliers with coefficients a is already scaled. Now, scale the next critical node, i.e., the inuts to the multiliers. In this case, the outut node. a) L -norm scaling. Determine the imulse resonse, by first determining the transfer function. a(z ) H(z) z b a b z a z b z a (b z ) n a b (b z ) n n or 0 for n < 0 h(n) a for n 0 a( /b) b n for n > 0 We get: S h(n) a + b which shall be a + b n b) L -norm scaling. Generally, it is difficult to find the maximal value of the magnitude function by analytical methods. However, in this simle case, (highass filter) we have max for z. Hence, H(e jωt ) a + b should be set to a + b 5.0 According to Jackson s lower bounds, Eqs.(5.0) (5.), the variance of round-off noise at the outut is bounded from below by a measure of the sensitivity. Hence, the sensitivity is low. Therefore, this filter can satisfy stringent requirements with short coefficient word length. 5. The inut is assumed to be scaled. The first critical node is after the first adder. The imulse resonse to this node is: 0.4, 0.4, 0, S 0.8

3 Increase the coefficients by a factor /S to 0.5 and 0.5, resectively. The imulse resonse to the outut node is (with the two first coefficients scaled): ( ), (0.5 ( 0.75)) + ( 0.5) 0.75), ( 0.5)( 0.75), 0, , 0.75, 0.375, 0, 0 S.5 Decrease the coefficients by multilying with a factor /S. We obtain the new values 0.5 and 0.5, resectively. The new scaled imulse resonse becomes: ( ), (0.5 ( 0.5)) + ( 0.5) 0.5), ( 0.5)( 0.5), 0 0.5, 0.5, 0.5, 0 Using L -norm scaling we get the transfer function to the first critical noise node: H(z) 0.4( z ). The maximal value of the magnitude function is gotten for z. Hence, H max 0.8. Thus, increase the first two coefficients to 0.5. The transfer function to the outut node is: H(z) 0.5( z ) 0.75( z ) The maximal value of the magnitude function is occur for z. Hence, H max.5. Thus, decrease the last two coefficients to 0.5. In this case, the two scaling criteria leads to the same coefficients. 5. a) The node v(n) must be scaled since it is inut (after the delay element) to the multilier with non-integer coefficient. Insert a scale coefficient, c, in front of the filter. The imulse resonse from the inut to the node is: h v (n) 0, n < 0 c b n, n 0 We get: S v h v (n) c b n c b 5.55c Now, let S v c The imulse resonse from the inut to the outut is: h(n) a 0 h v (n) + a h v (n ) 0, n < 0 a 0 c, n 0 a 0 c b n + a c b n, n

4 S y h(n) (a 0 c) + (a 0 c) [b n b n ] n (a 0 c) (b ) [ + b b n ] (a 0 c) [ + n (a 0 c) [ + b + b ] a Let S y a (b ) b b b )] b) All of the coefficients are non-integers. Hence, there are two noise sources to each "adder". The contribution from c and b is: σ y σ 0 h noise (n) σ 0 ( h(n) c ) σ y σ 0 c h(n) σ 0 c Since the filter is scaled. σ y σ 0 Q c The contribution from σ 0 and σ is: σ y Q σ y σ y + σ y Q where Q a) We have for a white stochastic sequence x(n) (which is a wideband signal): E{x(n)} 0 and the autocorrelation function is: r(nt) σ x δ(n). Hence, we have: S x (e jωt ) r(nt) e jωnt σ x n and S x max{ S x (e jωt ) } σ x b) For a narrow-band signal, for examle, a sinusoidal we have: S x A where A is the amlitude. 5.7 a) The critical overflow node that shall be scaled in direct form I is the outut node. The transfer function to this node is: H Critical (z) H AP (z) z + a z + b b z + a z + The L -norm of the magnitude function is:

5 H Critical (e jωt ) dωt H AP (e jωt ) dωt Hence, all of the oututs are roerly scaled and their scaling is indeendent of the L -norm. Of course, this is only valid for direct form I. For examle, the direct form II structure is not automatically scaled. b) The ordering of the sections does not effect the signal range since the transfer function from the inut of the allass filter to each of the critical nodes are allass functions. Also this fact is a secial case that is valid only for direct form I. 5.8 a) The variance of the quantization noise for System # is: σ Q / where Q 7 The variance of the quantization noise for System # is: σ Q / where Q 3 We assume that the average values are 0. The ower sectrum after the A/D converter is: S σ since the noise is white. A lowass filter with cutoff angle ω c T is laced after the converter in order to remove noise that should be aliased into the assband after the decimation. The variance of the noise after the lowass filter is: σ 3 r(0) 0 S e j0ωt dωt 0 S dωt ω c T 0 σ dωt σ f samle f σ f samle samle f samle (r(τ) is the autocorrelation function). We assume that the noise has zero mean. We must have: ω c T f samle f samle

6 so that the audio signal will not be attenuated by the lowass filter. If the quantization noise is to be less then or equal to the noise in System #, we must have: System σ Q ωt fsamle 44. khz σ Q System, after the A/D converter fsamle ωt H ω T c System ωt σ 3 ωt fsamle 44. khz σ 3 σ Hence, Q f samle f samle Q and f samle Q Q f samle 8 f samle ; f samle khz.64 MHz b) Decimation with a factor 04 yields a samle frequency of: f samle khz MHz Decimation can be done in stes of and the decimation filters can work at the lower samle frequency. The filter orders are almost the same for all stages. We need 0 decimation stages. The workload for one stage is only half of that of the receding stage. The workload of the whole decimation filter, counted from the outut, is:

7 N[ ] N 9 0 N 0 04 N where N is the workload of a single decimation filter working at the lowest samle frequency. c) System #: f samle f c f c System #: f samle f c 64 0 f c d) We assume that the accetable rile in the assband is about 0. db and the required stoband attenuation is about 50 db in both cases. We get: N System N System 5.9 Let E (e jωt ) be the error function caused by rounding the coefficients. We have: E(e jωt ) e jω(+)t [δh 0 + δh n cos(ωnt)] n but δh n Q c /. The deviation can be estimated in may ways, for examle, the maximal deviation or the variance of the deviation. Here we use an estimate of the maximal deviation: E(e jωt ) e jω(+)t [δh 0 + δh n cos(ωnt)] n δh 0 + δh n cos(ωnt) δh 0 + δh n cos(ωnt) n n E(e jωt ) δh 0 + δh n n Hence, E(e jωt ) Q N Q ( + ) (Bound #) Now, assume that the coefficients are randomly rounded and the errors are considered as indeendent random variables that are uniformly distributed in the interval [ Q/, Q/]. The variance is Q /. Let e 0 be the effective value of E(e jωt ) in the assband: e 0 f c N f c 0 E(e jωt ) dωt δh 0

8 The variance of e 0 is: σ E{ e 0 } N Q (Bound #) The required coefficient word length is estimated as follows. Let δ m be the accetable deviation in the assband of the stoband. Now, we must have: E(e jωt ) < δ m δ 0 where δ 0 is the deviation before quantization of the coefficients. Let the level of accetance of a coefficient set that does not fit the requirements be 5%. Hence the robability the coefficient set does not meet the secification is (assuming a normal distribution): P( E(e jωt ) σ x ) P( E(ejωT ) σ x ) 0.05 [ Φ( E(ejωT ) σ x )] 0.05 E(ejωT ) σ x E(e jωt ) σ x σ x δ m δ 0 N Q δ m δ 0 and Q (δ m δ 0 ) 3 N The number of bits that is required to reresent the coefficient deends on the largest coefficient. Hence. Q ( W c) [max{ [hn } ( W c) h0 ( W c) f s + f c f samle lowass filter. Hence, for a W c log { f samle f s + f c Q } log { f samle f s + f c (δ m δ 0 ) 3 N } W c log { f samle f s + f c (δ m δ 0 ) 3 N } For most lowass filter we have: N 3 f s f c f samle. W c log { f samle (f s + f c ) f samle f s f c (δ m δ 0 ) } In ractice, we may select δ m Min{ δ, δ } δ 0

9 W c log { f samle (f s + f c ) f samle f s f c δ m } W c log { f samle (f s + f c ) f samle f s f c } + log { Min{ δ, δ } } See also: Niedringshaus W.P., Steglitz., and odek D.: An Easily Comuted Performance Bound for Finite Wordlength Direct-Form FIR Digital Filters, IEEE Trans. on Circuits and Systems, Vol. CAS-9, No. 3,. 9-93, March 98.

The Noise Power Ratio - Theory and ADC Testing

The Noise Power Ratio - Theory and ADC Testing FH Irons, KJ Riley, and DM Hummels Abstract This aer develos theory behind the noise ower ratio (NPR) testing of ADCs. A mid-riser formulation is used for