NOISE IN A SPECTRUM ANALYZER. Carlo F.M. Carobbi and Fabio Ferrini Department of Information Engineering University of Florence, Italy

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1 NOISE IN A SPECTRUM ANALYZER by Carlo.M. Carobbi ad abio errii Departet of Iforatio Egieerig Uiversity of lorece, Italy 1. OBJECTIVE The objective is to easure the oise figure of a spectru aalyzer with differet ethods ad get failiar with oise ad its statistical properties.. EQUIPMENT You eed spectru aalyzer, a 5 Ω teriatio, a sigal geerator, a coaxial cable. I the figures we refer to a specific brad (hp) of spectru aalyzer. However, ay oder spectru aalyzer with digital play is adequate for the purpose of these experiets. 3. PROCEDURE Switch o the spectru aalyzer ad wait util war up is coplete (1 hour is sufficiet). After war up perfor selfcalibratio. The coect the 5 Ω teriatio to the iput of the spectru aalyzer. Use these settigs, which are coo to all the followig experiets: Ceter frequecy = 1 MHz, spa = Hz, iput atteuatio = db, resolutio badwidth ( 3 db badwidth, RBW) = 3 khz, video badwidth (VBW) = the largest available (e.g., 3 MHz). 3.1 Average level of the played oise i liear scale Set liear scale ad easureet uit icrovolt (μv). Select the referece level so that the oise fully occupies the vertical rage without exceedig the referece level (see ig. 1). Set saple detector. Narrow the video badwidth (or video average) so that oise appears as a sooth horizotal lie. Record the average oise level by usig the play lie ad calculate the oise factor through equatio (4). Use the provisioal value of the oise badwidth as give by B 1,1 3 khz 33 khz. A ore accurate value of the oise badwidth will be calculated at the last step, i 3.5. ially, calculate the oise figure N as N 1log 1 by usig (4) we obtai 5,8 ad N 4, db.. I the case represeted i ig. 1 we have 1,145 μv ad therefore,

2 ig. 1: Noise (blue trace, VBW = 3 MHz) ad average oise level (red trace, VBW = 1 Hz) i liear scale. 3. Crest factor of the played oise Use the sae settigs as i 3.1 ad set the video badwidth at its largest value. Take a sigle sweep ad, by usig the play lie, deterie the level L that is exceeded by oise peaks. To this purpose place the play lie at the vertical positio of the ( + 1)-th lower peak (see ig. ). The practical value of is coprised betwee 5 ad 1. Repeat the deteriatio of L at least te ties, by usig the sae value for, ad take the average value as the fial result. Calculate the probability p of exceedig the level L as p N where N is the total uber of played values (N is also called trace legth, i the case represeted i ig. we have N 41 ad 7 hece p,175 ). Calculate the crest factor C by usig equatio (9) ad the the oise factor by usig equatio (1) ad the provisioal value for B. ially calculate the oise figure N. See for exaple ig.. We calculate C,1 ad L,598 μv, therefore by usig equatio (1), we have 5.5 ad N 4, db.

3 ig. : Noise i liear scale (blue trace) ad play lie (red) at the eighth peak of oise. 3.3 Mea square value of the played oise Use the sae settigs as i 3.. Verify that the video badwidth is set to its largest value. Activate the arker ad place it ore or less at the ceter of the horizotal sweep (see ig. 3). Ru at least thirty sigle sweeps ad record each arker value. Calculate the ea square value (su of the squared values divided by the uber of values). The calculate the oise factor by usig equatio (14) ad the provisioal value for B. ially calculate the oise figure N. I the case 1 represeted i ig. 3 we obtaied (thirty sweeps) 1,73 1 (μv) ad therefore 7,7 ad N 4,3 db. ig. 3: Noise i liear scale. Marker captures oe rado value of oise. 3.4 Average level of the played oise i logarithic scale

4 Set logarithic scale i db uits ad 1 db per divisio. Set the referece level so that the oise is ostly over the botto divisio of the vertical scale. Set saple detector. Narrow the video badwidth (or video average) i order to sooth the oise, as show i ig. 4, ad easily idetify the average oise level by usig the arker or the play lie, see 3.1. Record the average oise level as ANL ad calculate the oise figure N by usig equatio (1) ad the provisioal value of the oise badwidth B. If, as i ig. 4, ANL 17,7 db the N 4, db. f ig. 4: Noise i logarithic scale (blue trace, VBW = 3 MHz) ad ANL (red trace, VBW = 1 Hz). 3.5 Equivalet oise badwidth Set liear scale ad easureet uit illivolt (V). Set peak detector. Set 1 db iteral atteuatio. Apply a 5 MHz siusoidal sigal at the iput of the spectru aalyzer. The aplitude of the siusoidal sigal is 1 V peak to peak. Verify that the resolutio badwidth is approxiately 3 khz (for exaple by usig the arker delta or siilar fuctio). Set the ceter frequecy so that the peak of the sigal is at the ceter of the horizotal sweep. Set the referece level so that the peak of the sigal touches the referece level (see ig. 5). Set the spa so that the tails of the bell-shaped respose of the iterediate-frequecy filter is utiguishable fro zero i the leftost ad rightost oe ad half divisio. Narrow the video badwidth i order to sooth the trace (if soe oise is visible i the tails of the respose of the filter). Record at least twety equally frequecy-spaced values H i of the filter respose ad the correspodig frequecies. Record the peak value of the respose of the filter H ad divide each recorded value by H, thus obtaiig hi Hi H. Calculate the square of each oralized value h ad the the quatity ( f is the frequecy spacig): i h1 h I 3... h h f. The result I is a approxiatio to the oise badwidth B (see 4.5) which ca be used i place of the 33 khz provisioal value. I the case represeted i ig. 5 it results B 33,1 khz, which is quite close to the provisioal 33 khz value. Itegratio over the full trace legth (41 values) affects oly the least sigificat figure (33, i place of 33,1 khz).

5 ig. 5: requecy respose (agitude) of the I filter. The oial RBW is 3 khz, the easured value is 9, khz. The easured oise badwidth (itegratio over values) is 33,1 khz. 4. THEORY The oise detected by a spectru aalyzer has a probability desity fuctio (pdf) give by where is the istataeous value of the oise, exp if f v v, (1) if v ktbr, is the oise factor of the spectru aalyzer, k is the 3 Boltza costat ( k 1,38 1 J/K), T is the absolute teperature (covetioally T T 9 K), B is the equivalet oise badwidth of the spectru aalyzer ad R 5 Ω. v ktbr is the equivalet oise root-ea-square (rs) voltage applied across the iput terials of the spectru aalyzer. The equivalet oise power at the iput of the spectru aalyzer is P ktb v R. The iteral atteuatio of the spectru aalyzer is assued to be db (i.e., A 1, where A is the power atteuatio i liear scale). If the iteral atteuatio is icreased the also the oise factor icreases (by the factor A, i.e. the oise factor becoes A ). The oise factor of the spectru aalyzer is usually (ad also here) evaluated by settig db iteral atteuatio. 4.1 Average level of the played oise i liear scale The average oise level i liear scale is obtaied as f d v, () where f is give by (1). Sice the played level is the iput level divided by to play to rs value of a equivalet siusoidal iput sigal), the the played average oise level is ad (the spectru aalyzer is calibrated v, (3)

6 4. (4) ktbr 4. Crest factor of the played oise The probability p that the oise detected by the spectru aalyzer exceeds the level L is give by Substitutig (1) ito (5) ad itegratig we have ad the Sice L L where We ca defie the crest factor ad L p f d. (5) L p exp v 1 L v l. (7) p L is the played level that is exceeded with probability p, the C of the played oise as C 1 v l L. (8) p L l 1 p. (9) v L. (1) ktbr C 1 (6) 4.3 Mea square value of the played oise The ea square value of the detected oise is give by Substitutig (1) ito (11) we obtai, solvig the itegral by parts, f d. (11). (1) v Sice the played levels are i ters of the rs value of the equivalet siusoidal sigal the ad v (13). (14) ktbr 4.4 Average level of the played oise i logarithic scale The average oise level i logarithic scale is obtaied as Substitutig (1) ito (15) we obtai log log f d. (15) 1 1 log1 log1 exp d v v. (16)

7 If the variable of itegratio is replaced by x v the we have where we used the fact that x log1 log1 x xexp dx log1 v, (17) f d 1. Nuerically coputig the itegral at the right side of (17) we obtai v log1,5 log1 (18) ad, takig to accout that the played level is expressed i ters of the rs value of a equivalet siusoidal iput, we have log 1 log 1 3,1,51 log 1 v dys. (19) The average oise level i logarithic scale is usually referred to with the acroy ANL ad is usually expressed i db, therefore fro (19) we obtai The oise figure of the receiver N is obtaied fro () as db ANL,51 ktb. () db N ANL,51 ktb. (1) 4.5 Equivalet oise badwidth The oise power P delivered to a 5 Ω load R, coected at the output of a two-port etwork, whose power gai is frequecy depedet ad give by G f, is P S f G f df () where S f is the oise spectral desity (expressed i W/Hz) delivered to the 5 Ω iput of the etwork. If the iput oise is white the it is costat with frequecy ad S f S, therefore P S G f df (3) urther, if the frequecy respose of the etwork is bell-shaped ad the axiu value of the gai over frequecy is G the where P SG B (4) 1 B G f df G (5) ad B is the equivalet oise badwidth of the etwork. The equivalet oise badwidth ca be also expressed i ters of the agitude of the voltage trasfer fuctio of the etwork H f. Ideed, sice G f H f obtai the, fro (5) we where H is the axiu value of H f. 1 B H f df H, (6) 5. CONCLUSION our differet ethods are here described to easure the oise figure of a spectru aalyzer. I order to obtai copatible results aog the ethods the followig recoedatios should be ipleeted. Noise is a rado process therefore

8 averagig over a greater saple size leads to ore precise results. Take this ito accout especially for the ethods described i 3. ad i 3.3. or exaple, the relative precisio of the estiate of the oise factor achievable by averagig thirty values is 1 3, i.e. about 18 % (or,7 db i ters of oise figure). urther, reeber that a spectru aalyzer is a physical device, ot a rado uber geerator based o a atheatical algorith. As a cosequece switchig istruet settigs (ceter frequecy, resolutio badwidth, atteuatio, referece level ) whe passig fro oe ethod to the other itroduces easureet iaccuracies. This is particularly true whe switchig the scale fro liear to logarithic, as required i order to pass fro the ethod described i 3.3 to that described i 3.4. I coclusio the agreeet aog the four oise figure values depeds o both the care with which the easureets are perfored ad o the quality of the spectru aalyzer. If experiets are perfored with the due care ad a oder spectru aalyzer is used the a agreeet of up to 1 db is absolutely acceptable, while ore tha db is questioable. 6. REERENCES [1] Keysight Techologies, Applicatio Note 15, Spectru Aalysis Basics, 15. [] Agilet Techologies, Applicatio Note 133, Spectru Aalyzer Measureets ad Noise, 3.