Narrow-Band FMand Wide-Band FM Presented By N.A.PAPPATHI AP / ECE, NPRCET. Carrier Resting fc Increasing fc Decreasing fc Increasing fc Resting fc Modulating signal FM • Angle Modulation Types – Frequency Modulation 1. NarrowBand FM (NBFM) 2. WideBand FM (WBFM) – Phase Modulation is small compared to one radian • For WBFM.FM Equation • The FM signal expression is given by s(t ) Ac cos[2f ct sin(2f mt )] • On expanding using cos A B cos A cos B sin A sin B s(t ) Ac cos(2f ct ) cos[ sin(2f mt )] Ac sin(2f ct ) sin[ sin(2f mt )] • For NBFM. is large compared to one radian . Narrow-Band Frequency Modulation For small values of . the expression for FM signal will be simplified to s(t ) Ac cos(2f ct ) Ac sin(2f ct )sin(2f mt ) . cos( sin( 2f mt )) 1 sin( sin( 2f mt )) sin( 2f mt ) Thus for NBFM. NBFM Generation – Block Diagram . can Thenbe expanded as s(t ) Ac cos(2f ct ) 1 Ac cos[2 ( f c f m )t ] cos[2 ( f c f m )t ] 2 since 1 sin A sin B cos A B cos A B 2 When comparing the s(t) for FM with the s(t) equation of AM given below. S (t ) Ac cos(2f ct ) 1 Ac cos[2 ( f c f m )t ] cos[2 ( f c f m )t ] AM 2 • We came to know that the basic difference between AM signal and NBFM signal is that the algebraic sign of Lower side frequency in NBFM is reversed • Hence the BW of NBFM is same as that of AM i.e 2fm . Spectrum of NBFM (single-sided plot) . s(t ) Ac cos[2f ct sin(2f mt )] • Complex representation of BandPass signals is used to describe the s(t) equation of FM. s(t ) Re[ Ac exp( j 2f ct j sin(2f mt ))] s(t ) Re[ s (t ) exp(Where j 2f ct ](t) = Phase deviation where s (t ) Ac exp[ j sin(2f mt )] . the below general FM equation is analyzed.Wide-Band Frequency Modulation WIDE-BAND FREQUENCY MODULATION • For arbitrary value of modulation index β. Unlike FM signal s(t). the complex envelope s (t ) is a periodic function with fundamental frequency fm and so can be expanded in Complex Fourier series as s(t ) cn exp( j 2nf mt ) where the Fourier coefficient cn is given by cn f m f m Ac 1/ 2 f 1/ 2 f 1/ 2 f s(t ) exp( j 2nf mt )dt m m m 1/ 2 f exp[ j sin(2f mt ) j 2nf mt ]dt m . the cn can be written as Ac cn exp[ j ( sin x nx)]dx 2 The integral on the right hand side is a function of “n” and and is known as the Bessel function of order n and argument . Bessel function is conventionally denoted by Jn( ) .Defining a new variable x=2Пfmt. we get s(t ) Ac Re J n ( )exp[ j 2 ( f c nf m )t ] n . we get s(t ) Ac J n ( )exp( j 2nf mt ) Substituting complex envelope in s(t). J n ( ) 1 exp[ j( sin x nx)]dx 2 Thus cn Ac J n ( ) Substituting cn value in complex envelope.Therefore it follows that. therefore obtained by taking FT of both sides Ac S( f ) J ( ) ( f f nf ) ( f f nf ) n c m c m 2 . s(t ) Ac Re J n ( )exp[ j 2 ( f c nf m )t ] n The discrete spectrum of s(t) is. Jn () = (-1)n J-n () .Properties of Bessel Functions Property .1: For n even. we have Jn () = (-1) J-n () Thus. we have Jn () = J-n () For n odd. Property .3: 2 J ( ) 1 n n .2: For small values of the modulation index we have J0 () 1 J1 () /2 J3 () 0 for n > 2 Property . TABLE OF BESSEL FUNCTIONS . β Vs Jn(β) . β Vs Jn(β) . AMPLITUDE SPECTRUM . . unlike AM. For small values of .… 2.Observations from WBFM and Bessel functions 1. This corresponds to NBFM 3. 2fm. only J0() and J1() have significant values . the amplitude of carrier component is dependent on . 3fm. . That is. Spectrum of FM contains fc and infinite set of side frequencies located symmetrically on either side of fc at separations of fm. The amplitude of carrier component varies with according to J0(). and so FM signal is effectively composed of a carrier and a single pair of side frequencies at fc ± fm. • Angle modulation is capable of handing a greater dynamic range of modulating signal without distortion than AM. • Angle modulation is resistant to propagation-induced selective fading since amplitude variations are unimportant and are removed at the receiver using a limiting circuit.Advantages • Wideband FM gives significant improvement in the SNR at the output of the RX which proportional to the square of modulation index. • Angle modulation allows the use of more efficient transmitter power in information. • Angle modulation is very effective in rejecting interference. (minimizes the effect of noise). . • Angle modulation requires more complex and expensive circuits than AM.Disadvantages • Angle modulation requires a transmission bandwidth much larger than the message signal bandwidth. . Power in Angle-Modulated Signal The power in an angle-modulated signal is easily computed n 2 2 1 P Ac J n ( ) 2 n . the bandwidth of an anglemodulated signal can be defined by considering only those terms that contain significant power. However.Transmission Bandwidth of FM signals Theoretically. since the values of Jn() become negligible for sufficiently large n. a FM signal contains an infinite number of side frequencies so that the bandwidth required to transmit such signal is infinite. . the bandwidth of a FM signal can be determined by knowing the modulation index and using the Bessel function table. .In practice. .Example: Determine bandwidth with table of Bessel functions Calculate the bandwidth occupied by a FM signal with a modulation index of 2 and a highest modulating frequency of 2.5 kHz. Referring to the table. . The bandwidth can then be determined with the simple formula B. we can see that this produces six significant pairs of sidebands.W 2nf max where n is the number of significant sidebands. the bandwidth of the FM signal is B. 2 6 2.Using the example above and assuming a highest modulating frequency of 2.5 kHz.5 30kHz .W . we can find that the FM signal is effectively limited to finite number of significant side frequencies. • Hence Effective Bandwidth is used in which the side frequencies with significant power only is considered. • In practice. • Another method is to approximate the Transmission Bandwidth for small value of β using Carson’s rule 1 BT 2f 2 f m 2f 1 .Transmission Bandwidth of FM Signals • Since FM has infinite number of side frequencies. BW required to transmit such a signal is infinite for the ideal case. 2(2. the bandwidth would be B.5kHz 15kHz .5kHz 5kHz ) 2 7.5 kHz.W .Example: Assuming a maximum frequency deviation of 5 kHz and a maximum modulating frequency of 2. 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