EE (082) Chapter IV: Angle Modulation Lecture 19 Dr. Wajih Abu-Al-Saud

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1 EE (082) Chaer IV: Angle Modulaion Lecure 19 Dr. Wajih Abu-Al-Saud Wideband and PM For he signal shown below, he value o k ay no saisy he condiion k 1, and hereore he aroxiaion used in narrowband ay no be alicable. For signals g () = A cos ωc + k ( α) dα. ha do no saisy k 1, couing he bandwidh is ore diicul han or narrowband signals. To coue he bandwidh o wideband, we will aroxiae he signal g () by anoher signal ha resuls ro odulaing a saled version o he essage signal. Tha is, insead o using he signal () as he essage signal or odulaing he carrier, we will use an aroxiaion ˆ () ha is obained by saling () as shown in he ollowing igure. () ˆ () The value o ha we will use is he axiu allowable value ha will insure ha he () can be reconsruced ro ˆ () wihou loss o inoraion. This axiu value or is obained using he Nyquis saling heore, which saes ha or a signal () wih a bandwidh o B (Hz), he iniu saling requency is 2B. Thereore, he axiu or is given by = 1/2B. O course a saller value or will be beer since i gives a beer aroxiaion or () bu is unnecessary. So, o ind he aroxiae bandwidh o signal, le us assue ha he original essage signal () is bounded in aliude by he wo values and. Thereore, (). Then, we can sraigh orward say ha ˆ ().

2 EE (082) Chaer IV: Angle Modulaion Lecure 19 Dr. Wajih Abu-Al-Saud Now, he aroxiae signal gˆ () o g () is obained by odulaing ˆ () as gˆ () cos ˆ = A ωc + k ( α) dα Since ˆ () is consan over eriods o = 1/2B, he insananeous requency o gˆ () will be consan over eriods o = 1/2B. The signal gˆ () will look like as ollows. () ˆ () gˆ () Since ˆ (), he insananeous requency o gˆ () (and also g () ) will be in he range ω k ω () ω + k. c i c This eans ha he insananeous requency changes over a range o ω = k (his can also be wrien as = k /2π) on each side around he carrier requency ω c. So, o aroxiae he bandwidh o he o original signal g (), we will coue he aroxiae bandwidh o he aroxiaion signal gˆ () by inding is requency secru. Since gˆ () is coosed o blocks o sinusoids wih dieren requencies ha are in he range o requencies o ωc k ωi() ωc + k, we can ind he secru o each o hese blocks indeendenly and hen add hese secrus o ge he overall secru o gˆ ().

3 EE (082) Chaer IV: Angle Modulaion Lecure 19 Dr. Wajih Abu-Al-Saud Consider he ar o gˆ () ha shown below The signal z() is given by ( ) ( ω ) z () = A rec 2B cos 0 i, where ω k ω () ω + k. The Fourier ransor o z() is c i c Aπ ω ω i ω+ ω Z ( ω) = sinc e + sinc e 4B 4B 4B ( ω ω ) ( ω ω ) j i 0 i j + i 0. Reebering ha he sinc uncion looks like he ollowing Skeching he agniude o Z(ω) (i.e., agniude secru o z()) will give he ollowing (since he colex exonenials have a agniude o one and he wo coonens shied o he le and righ alos do no inerere wih each oher)

4 EE (082) Chaer IV: Angle Modulaion Lecure 19 Dr. Wajih Abu-Al-Saud Adding he secru o he dieren signals like z() given above will give us he secru o he aroxiaion signal gˆ () Gˆ ( ω) I we assue ha he sidebands (he sall hus a he wo edges o a sinc uncion) have negligible ower, and knowing ha ω = k, we see ha he bandwidh o an signal is aroxiaely equal o 2k + 8 π B (rad/s). 2 ω+ 8 πb (rad/s) Using he ac ha = k /2π, he bandwidh in Hz becoes 2k + 4 B (Hz) 2π B (Hz) [ B ] (Hz) In racice, his bandwidh is higher han he acual bandwidh o signals. Consider or exale narrowband. Using his orula or he bandwidh, we see ha he bandwidh is wice he acual bandwidh. In ac, a ore accurae relaionshi is known as CARSON s Rule, which is given by where and [ ω πb ] [ ] B (Hz) 2 + B (Hz), (rad/s) B = Bandwidh o he Message Signal () in Hz, ω = k = k /2π.

5 EE (082) Chaer IV: Angle Modulaion Lecure 19 Dr. Wajih Abu-Al-Saud We will deine a quaniy β such ha ω k β = = B B This is known as he MODULATION INDEX o he signal. Exercise: A sinusoidal essage signal () wih a requency o 5000 Hz and aliude o 5 V is used o generae an signal wih aliude A = 10 V using a odulaor wih k = 2π 3000 and a carrier requency o 5 MHz. Find he ollowing: a) The requency deviaion and he range o insananeous requency (ind boh in rad/s and in Hz) or he signal. b) The odulaion index o he signal. c) The Bandwidh o he signal (boh in rad/s and Hz).