Chap 6 Noise in Analog Modulation Wireless Information Transmission System Lab. Institute of Communications Engineering g g National Sun YatYat-sen University 1 Outline ◊ 6.1 Introduction ◊ 6.2 Receiver Model ◊ 63 N 6.3 Noise i iin DSB-SC DSB SC R Receivers i ◊ 6.4 Noise in AM Receivers ◊ 6.5 Noise in FM Receivers 2 Yu-sing Lin (林育星) Chap 6.1 Introduction Wireless Information Transmission System Lab. Institute of Communications Engineering g g National Sun YatYat-sen University 3 6.1 Introduction ◊ To undertake an analysis of noise in continuous-wave (CM) modulation systems, we need a receiver model. ◊ The customary practice is to model the receiver noise (channel noise) as additive, white, and Gaussian. These simplifying assumptions enable us to obtain a basic understanding of the way in which noise affects the performance f off th the receiver. i 4 Yu-sing Lin (林育星) Chap 6.2 Receiver Model Wireless Information Transmission System Lab. Institute of Communications Engineering g g National Sun YatYat-sen University 5 6.2 Receiver Model ◊ ◊ ◊ ◊ s(t) denotes the incoming modulated signal. w(t) denotes front-end receiver noise. The power spectral density of the noise w(t) is denoted by N0/2, defined for both positive and negative frequencies. N0 is the average noise power per unit b d idth measuredd att the bandwidth th front f t endd off the th receiver. i The bandwidth of this band-pass filter is just wide enough to pass the modulated signal without distortion. distortion Assume the band-pass filter is ideal, having a bandwidth equal to the transmission bandwidth BT of the modulated signal s(t) , and a mid midband frequency equal to the carrier frequency fc , fc >> BT . 6 Yu-sing Lin (林育星) 6.2 Receiver Model Idealized characteristic of band-pass filtered noise. ◊ The filtered noise n(t) may be treated as a narrow band noise represented in the canonical form: n ( t ) = nI ( t ) cos ( 2πf c t ) − nQ ( t ) sin ( 2πf c t ) ( 6.1) where nI((t)) is the in-phase p noise component p and nQ((t)) is the quadrature noise component, both measured with respect to the carrier wave Accos(2πfct). Yu-sing Lin (林育星) 7 6.2 Receiver Model ◊ The filtered signal x(t) available for demodulation is defined by x (t ) = s (t ) + n (t ) ◊ ◊ The average noise power at the demodulator input is equal to the total area under the curve of the power spectral density SN( f ): N0 Pavg-noise = 2 × BT × = BT N 0 2 Input signal-to-noise ratio (SNR)I is defined as: ( SNR ) I ◊ ( 6.2 ) = average power of the modulated signal s ( t ) average power of the filtered noise n ( t ) Output signal-to-noise ratio (SNR)O is defined as: ( SNR )O = average power of the demodulated message signal average power of the noise 8 measured at the receiver output Yu-sing Lin (林育星) 6.2 Receiver Model ◊ Channel signal-to-noise ratio ( SNR )C ◊ = average power of the modulated signal average power of noise in the message BW measured at the receiver input This ratio Thi i may be b viewed i d as the h signal-to-noise i l i ratio i that h results l from baseband (direct) transmission of the message signal m(t) without modulation, modulation as demonstrated in the following figure: ◊ ◊ The message power at the low-pass filter input is adjusted to be the same as the average power of the modulated signal The low-pass filter passes the message signal and rejects out-of-band noise. 9 Yu-sing Lin (林育星) 6.2 Receiver Model ◊ Figure of merit ◊ ◊ ◊ For the purpose of comparing different continuous-wave (CW) modulation systems, we normalize the receiver performance by dividing the output signal-to-noise signal to noise ratio by the channel signal signal-to-noise to noise ratio. The higher the value of the figure of merit, the better will the noise performance of the receiver be. The figure of merit may equal one, be less than one, or be greater than one, depending on the type of modulation used. ( SNR )O Figure of merit= ( SNR )C 10 Yu-sing Lin (林育星) Chap 6.3 Noise in DSB DSB--SC Receiver Wireless Information Transmission System Lab. Institute of Communications Engineering g g National Sun YatYat-sen University 11 6.3 Noise in DSBDSB-SC Receiver ◊ The model of a DSB-SC receiver using a coherent detector Figure g 6.4 ◊ ◊ The amplitude of the locally generated sinusoidal wave is assumed to be unity unity. For the demodulation scheme to operate satisfactorily, it is necessary ecessa y tthat at tthee local oca osc oscillator ato be synchronized sy c o ed both bot in phase p ase and in frequency with the oscillator generating the carrier wave in the transmitter. We assume that this synchronization has been achieved. 12 Yu-sing Lin (林育星) 6.3 Noise in DSBDSB-SC Receiver ◊ ◊ ◊ The DSB-SC component of the modulated signal s(t) is expressed as s ( t ) = CAc cos ( 2π f ct ) m ( t ) ( 6.4 ) where C is the system dependent scaling factor factor. The purpose of which is to ensure that the signal component s(t) is measured in the same units as the additive noise component n(t). m(t) is the sample function of a stationary process of zero mean, whose power spectral density SM( f ) is limited to a maximum frequency W, i.e. W is the message bandwidth. The average power P of the message signal is the total area under th curve off power spectral the t l density d it W P = ∫ S M ( f ) df −W 13 Yu-sing Lin (林育星) 6.3 Noise in DSBDSB-SC Receiver ◊ In Example 5.12: Mixing of a Random Process with a Sinusoidal Process Y ( f ) = X ( t ) cos ( 2π f c t + Θ ) RY (τ ) = ◊ ◊ ◊ 1 RX (τ ) cos ( 2π f cτ ) 2 1 SY ( f ) = ⎡⎣ S X ( f − f c ) + S X ( f + f c ) ⎤⎦ 4 Therefore, the average power of the DSB-SC modulated signal component s(t) is given by: C 2 Ac2 P 2 The average power of the noise in the message BW is WN0 The channel signal-to-noise ratio of the DSB-SC modulation system is: C 2 Ac2 P (SNR )C ,DSB = 2WN 0 14 Yu-sing Lin (林育星) 6.3 Noise in DSBDSB-SC Receiver ◊ ◊ Next, we wish to determine the output signal-to-noise ratio. Usingg the narrowband representation p of the filtered noise n(t), ( ), the total signal at the coherent detector input may be expressed as: x (t ) = s (t ) + n (t ) = CAc cos ( 2π f c t ) m ( t ) + nI ( t ) cos ( 2π f c t ) − nQ ( t ) sin ( 2π f c t ) ◊ ( 6.7 ) The output of the product-modulator product modulator component of the coherent detector is: υ ( t ) = x ( t ) cos ( 2π f c t ) cos(α+β) + cos(α−β) = 2cos(α)cos (β) 1 1 sin(α+β) + sin(α−β) = 2sin(α)cos(β) = CAc m ( t ) + nI ( t ) 2 2 1 1 + ⎡⎣CAc m ( t ) + nI ( t ) ⎤⎦ cos ( 4π f c t ) − Ac nQ ( t ) sin ( 4π f c t ) 2 2 low-pass p filter,, BW=W 1 1 y ( t ) = CAc m ( t ) + nI ( t ) 2 2 ( 6.8 ) 15 Yu-sing Lin (林育星) 6.3 Noise in DSBDSB-SC Receiver ◊ ◊ ◊ Equation (6.8) indicates the following: ◊ The Th message signal i l m(t) (t) andd in-phase i h noise i componentt nI(t) off the filtered noise n(t) appear additively at the receiver output. ◊ The quadrature component nQ(t) of the noise n(t) is completely rejected by the coherent detector. We note that these two results are independent of the input signalto-noise ratio. Thus, coherent detection distinguishes itself from other demodulation techniques in the important property: the output message component is unmutilated and the noise component always appears additively with the message, irrespective of the input signal-to-noise ratio. 16 Yu-sing Lin (林育星) 6.3 Noise in DSBDSB-SC Receiver ◊ 1 1 The receiver output signal : y ( t ) = CAc m ( t ) + nI ( t ) (6.8) 2 2 The average power of message component may be expressed as ◊ C 2 Ac2 P Pavg = 4 Th average power off the The th noise i att the th receiver i output t t is i ◊ 2 1 ⎛1⎞ ⎜ ⎟ 2WN 0 = WN 0 2 ⎝2⎠ ◊ ◊ In the case of DSB-SC modulation, the band-pass filter in Figure 6.4 has a band-width band width BT equal to 2W in order to accommodate the upper and lower sidebands of the modulated signal s(t). Therefore, the average power of the filtered noise n(t) is 2WN0. F From P Property t 5 off S Sectt 55.11, 11 th the average power off th the (low-pass) (l ) nI(t) iis the same as that of the (band-pass) filtered noise n(t). 17 Yu-sing Lin (林育星) 6.3 Noise in DSBDSB-SC Receiver ◊ ◊ The output signal-noise ratio for DSB-SC C 2 Ac2 P 4 (SNR )O ,DSB-SC = WN 0 2 C 2 Ac2 P = 2WN 0 We obtain the figure of merit (SNR )O (SNR )C ( 6.9 ) ( 6.10 ) =1 DSB-SC 18 Yu-sing Lin (林育星) Chap 6.4 Noise in AM Receivers Wireless Information Transmission System Lab. Institute of Communications Engineering g g National Sun YatYat-sen University 19 6.4 Noise in AM Receivers ◊ A full AM signal is given by s ( t ) = Ac ⎡⎣1 + ka m ( t ) ⎦⎤ cos ( 2πf c t ) ◊ ( 6.11) where Accos(2πfct) is the carrier wave, m(t) is the message signal and bandwidth is W, ka is a constant that determines the percentage modulation. We would like to perform noise analysis for an AM system using an envelope l detector. d 20 Yu-sing Lin (林育星) 6.4 Noise in AM Receivers ◊ ◊ We perform the noise analysis of the AM receiver by first determining the channel signal-to-noise signal to noise ratio, ratio and then the output signal-to-noise ratio. We can easily obtain average power of the AM signal s ( t ) = Ac cos ( 2πf c t ) + Ac ka m ( t ) cos ( 2πf c t ) 1 2 1 2 2 Ac + Ac ka P 2 2 The average power of noise in the message bandwdith is WN0 (the same as the DSB-SC system) The channel signal signal-to-noise to noise ratio for AM is therefore: Ps = ◊ ◊ (SNR )C ,AM = 21 Ac2 (1 + ka2 P ) 2WN 0 Yu-sing Lin (林育星) 6.4 Noise in AM Receivers ◊ The filtered signal x(t) applied to the envelope detector in the receiver is ggiven by: y x (t ) = s (t ) + n (t ) = ⎡⎣ Ac + Ac ka m ( t ) + n1 ( t ) ⎤⎦ cos ( 2πf c t ) − nQ ( t ) sin ( 2πf c t ) y ( t ) = envelope of x ( t ) { } = ⎡⎣ Ac + Ac ka m ( t ) + nI ( t ) ⎤⎦ + n ( t ) 2 2 Q 12 6 13) ( 6.13 ( 6.14 ) Assume average g carrier power p >> average g noise power p y (t ) ◊ ◊ Ac + Ac ka m ( t ) + nI ( t ) ( 6.15) The dc term or constant term Ac may be removed simply by means of a blocking capacitor. If we ignore the dc term Ac, we find that the remainder has a form similar to the output of a DSB-SC receiver using coherent detection. 22 1 1 y ( t ) = CAc m ( t ) + nI ( t ) 2 2 ( 6.8) Yu-sing Lin (林育星) 6.4 Noise in AM Receivers ◊ The output signal-to-noise ratio of an AM using an envelope detector is approximately (SNR )O ,AM ◊ Ac2 ka2 P 2WN 0 ( 6.16 ) Eq. (16) is valid only if the following two conditions are satisfied ◊ The average noise power is small compared to the average carrier power at the envelope detector input. ◊ The amplitude sensitive ka is adjusted for a percentage modulation less th or equall to than t 100 percent. t (| ka m(t)|<=1) )|< 1) 23 Yu-sing Lin (林育星) 6.4 Noise in AM Receivers ◊ Comparison of figure of merit ( AM, DSB-SC, SSB ) (SNR )O (SNR )C ◊ ◊ ◊ AM using Envelope Detector (SNR )O (SNR )C ka2 P ≤1 2 1 + ka P DSB-SC (SNR )O = (SNR )C =1 SSB The figure g of merit of a DSB-SC receiver or that of an SSB receiver using coherent detection is always unity. The corresponding figure of merit of an AM receiver using envelope detection is always less than unity. In other words,, the noise pperformance of an AM receiver is always inferior to that a DSB-SC receiver. This is due to the wastage of transmitter power, which results from transmitting the carrier i as a component off AM wave. 24 Yu-sing Lin (林育星) 6.4 Noise in AM Receivers ◊ Example 6.1 Single-Tone Modulation ◊ ◊ Consider a sinusoidal wave of frequency fm and amplitude Am as the modulating wave, as shown by m(t) = Amcos(2πfmt) The corresponding AM wave is s(t) = Ac [1+ μcos(2πfmt)] cos(2πfct) modulation factor : μ = kaAm ◊ The average power of the modulation wave m(t) is (assuming a load resistor of 1ohm) 1 P = Am2 2 25 Yu-sing Lin (林育星) 6.4 Noise in AM Receivers ◊ We obtain the figure of merit (SNR )O (SNR )C ◊ ◊ AM 1 2 2 ka Am 2 μ = 2 = 1 2 2 2 + μ2 1 + ka Am 2 6 18 ) ( 6.18 When μ = 1 (100% modulation using envelope detection), we get a figure of merit = 1/3. This means that, other factors being equal, an AM system (using envelope detection) must transmit three times as much average power as a suppressed-carrier d i system (using ( i coherent h detection) d i ) in i order to achieve the same quality of noise performance. 26 Yu-sing Lin (林育星) Chap 6.5 Noise in FM Receivers Wireless Information Transmission System Lab. Institute of Communications Engineering g g National Sun YatYat-sen University 27 6.5 Noise in FM Receivers ◊ The receiver model is given by: ◊ The noise w(t) is modeled as white Gaussian noise of zero mean and power spectral density N0/2. /2 The received FM signal s(t) has a carrier frequency fc and transmission bandwidth BT, such that only a negligible amount of power lies outside the frequency band fc ± BT /2 for positive frequencies. The band-pass filter has a mid-band frequency fc and bandwidth BT and th f therefore passes the th FM signal i l essentially ti ll without ith t distortion. di t ti Ordinary, BT is small compared with the mid-band frequency fc so that we may use the narrowband representation for n(t), the filtered version of receiver noise w(t), in terms of its in-phase and quadrature components. Yu-sing Lin (林育星) 28 ◊ ◊ ◊ 6.5 Noise in FM Receivers ◊ ◊ ◊ In an FM system, the message information is transmitted by variations i i off the h instantaneous i frequency f off a sinusoidal i id l carrier i wave, and its amplitude is maintained constant. A variations Any i ti off the th carrier i amplitude lit d att the th receiver i input i t mustt result from noise or interference. The limiter is used to remove amplitude variations by clipping the modulated wave at the filter output almost to the zero axis. ◊ The resulting rectangular wave is rounded off by another bandpass filter that is an integral part of the limiter, thereby suppressing harmonics of the carrier frequency. ◊ The filter output is again sinusoidal, with an amplitude that is practically ti ll independent i d d t off the th carrier i amplitude lit d att the th receiver i input. 29 Yu-sing Lin (林育星) 6.5 Noise in FM Receivers ◊ The discriminator consists of two components: ◊ ◊ ◊ ◊ A slope network or differentiator with a purely imaginary transfer function that varies linearly with frequency. It produces a hybrid-modulated wave in which both amplitude p and frequency q y varyy in accordance with the message g signal. An envelope detector that recovers the amplitude variation and thus reproduces the message signal. signal The slope network and envelope detector are usually implemented as integral parts of a single physical unit. The post-detection filter, labeled “baseband low-pass filter,” has a bandwidth that is just large enough to accommodate the highest f frequency component off the h message signal. i l ◊ This filter removes the out-of-band components of the noise at th discriminator the di i i t output t t andd thereby th b keeps k the th effect ff t off the th output noise to a minimum. 30 Yu-sing Lin (林育星) 6.5 Noise in FM Receivers ◊ The filtered noise at the band-pass filter output is defined as: n ( t ) = nI ( t ) cos ( 2π f c t ) − nQ ( t ) sin ( 2π f c t ) n ( t ) = r ( t ) cos ⎡⎣ 2πf c t +ψ ( t ) ⎤⎦ ( 6.21) r ( t ) = ⎡⎣ n ( t ) + n ( t ) ⎤⎦ ∼ Rayleigh distribution ⎡ nQ ( t ) ⎤ −1 ψ ( t ) = tan ⎢ ∼ U ( 0,2π ) ⎥ ⎣ nI ( t ) ⎦ The incoming FM signal s(t) is given by ( 6.22 ) 2 I ◊ 2 Q 12 t ⎡ s ( t ) = Ac cos 2πf c t + 2πk f ∫ m (τ ) dτ ⎤ ⎢⎣ ⎥⎦ 0 t ( 6.23) ( 6.24 ) φ ( t ) = 2πk f ∫ m (τ ) dτ ( 6.25) s ( t ) = Ac cos ⎡⎣ 2πf c t + φ ( t ) ⎦⎤ ( 6.26 ) 0 31 Yu-sing Lin (林育星) 6.5 Noise in FM Receivers ◊ The noisy signal at the band-pass filter output is: x ( t ) = s ( t ) + n ( t ) = Ac cos ⎡⎣ 2πf c t + φ ( t ) ⎤⎦ + r ( t ) cos ⎡⎣ 2πf c t +ψ ( t ) ⎤⎦ ⎧⎪ r ( t ) sin ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦ ⎫⎪ θ ( t ) = φ ( t ) + tan ⎨ ⎬ ⎡ ⎤ A + r t cos ψ t − φ t ( ) ⎣ ( ) ( )⎦ ⎪⎭ ⎪⎩ c The envelope of x(t) is of no interest to us, because any envelope variations at the band-pass band pass output are removed by the limiter. limiter Our motivation is to determine the error in the instantaneous frequency of the carrier wave caused by the presence of the filtered noise n(t). −1 ◊ ◊ 32 Yu-sing Lin (林育星) 6.5 Noise in FM Receivers ◊ ◊ With the discriminator assumed ideal, its output is proportional to θ’(t)/2 ( ) π where θ’(t) ( ) is the derivative of θ(t) ( ) with respect p to time. We need to make certain simplifying approximations so that our y may y yield y useful results. analysis ⎧⎪ r ( t ) sin ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦ ⎫⎪ θ ( t ) = φ ( t ) + tan ⎨ ⎬ ⎪⎩ Ac + r ( t ) cos ⎡⎣ψ ( t ) − φ ( t ) ⎦⎤ ⎪⎭ (1) when CNR 1, A r ( t ) r (t ) r ( t ) sin ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦ θ (t ) φ (t ) + sin ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦ Ac A + r ( t ) cos ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦ −1 c t r (t ) 0 Ac θ ( t ) 2π k f ∫ m ( t )dt + 1 dθ ( t ) υ (t ) = 2π dt c sin ⎡⎣ψ ( t ) − φ ( t ) ⎤⎦ ∼ ( 2) x k f m ( t ) + nd ( t ) 33 r ( t ) sin ⎣⎡ψ ( t ) − φ ( t ) ⎦⎤ Ac 1, tan −1 ( x ) ≈ x (6.31) Yu-sing Lin (林育星) 6.5 Noise in FM Receivers { } 1 d nd ( t ) = r ( t ) sin i ⎡⎣ ψ ( t ) − φ ( t ) ⎦⎤ 2π Ac dt nd ( t ) nd ( t ) ◊ { } 1 d r ( t ) sin ⎡⎣ ψ ( t ) ⎤⎦ 2π Ac dt 1 dnQ ( t ) 2π Ac dt ψ ( t ) ∼ U ( 0,2π ) ⇒ Assume ψ ( t ) − φ ( t ) ∼ U ( 0,2π ) If such an assumption were true, then the noise at the discriminator output would be indep. of the modulating signal and would depend only on the characteristics of the carrier and narrowband noise. This assumption is justified provided that the carrier-to-noise ratio is high. nQ(t) : quadrature component This means that the additive noise nd(t) appearing at the discriminator output is determined effectively by the carrier amplitude Ac and the quadrature component nQ(t) of the narrowband noise n(t). 34 Yu-sing Lin (林育星) 6.5 Noise in FM Receivers ◊ The output signal-to-noise ratio is defined as the ratio of the g output p signal g power p to the average g output p noise ppower. average ◊ From Eq. (6.31), the message component in the discriminator output, andd therefore h f the h low-pass l fil output, is filter i kf m(t). () The average output signal power is equal to kf 2P, where P is the average power off th the message signal i l m(t). (t) ◊ ◊ ◊ To determine the average output noise power, we note that the noise nd(t) at the discriminator output is proportional to the time derivative of the quadrature noise component nQ(t). The differentiation of a function respect to time corresponds to multiplication of its Fourier transform by j2πf. We may obtain the noise i nd(t) ( ) bby passing i nQ(t) ( ) though h h a li linear filter fil with i h a transfer f j 2π f jf = . function: 2π Ac Ac 35 Yu-sing Lin (林育星) 6.5 Noise in FM Receivers d x ( t ) ⎯⎯ → j 2π fX ( f ) dt 2 ( 2 ) SY ( f ) = H ( f ) S X ( f ) (1) f2 S N d ( f ) = 2 S NQ ( f ) Ac ⎧ N0 f 2 ⎪ 2 , S Nd ( f ) = ⎨ Ac ⎪ 0, ⎩ ⎧ N0 f 2 ⎪ 2 , S NO ( f ) = ⎨ Ac ⎪ 0, ⎩ 0 ◊ f ≤ BT 2 X(t) () H( f ) Y(t) () otherwise f ≤W otherwise For wide-band FM, we usually find that W is smaller ll than h BT /2, /2 where h BT is i the h transmission i i bandwidth of the FM signal. This means that out-of-band components of noise no(t) will be rejected. 36 Yu-sing Lin (林育星) 6.5 Noise in FM Receivers ◊ ◊ ◊ ◊ In FM system, increasing the carrier power has a noise-quieting effect. N0 W 2 2 N 0W 3 2 Average power of output noise = 2 ∫ f df = ∝ 2 2 − W Ac Ac 3 Ac k 2f P 3 Ac2 k 2f P = ( 6.40 ) We obtain ( SNR )O ,FM = 3 2 3 2 N 0W 3 Ac 2 N 0W The average power in the modulated signal s(t) is A2c /2, and the average noise power in the message bandwidth is WNo. The channel signal to noise ratio (SNR)C,FM is Ac2 (SNR )C ,FM = ( 6.41) 2WN 0 2WN Figure of merit for frequency modulation: 3k 2f P (SNR )O = 6 42 ) ( 6.42 2 (SNR )C FM W 37 Yu-sing Lin (林育星) 6.5 Noise in FM Receivers ◊ Ex 6.2 Single-Tone Modulation ◊ A sinusoidal wave of frequency fm as the modulating signal, and assume a peak frequency deviation ∆f . The FM signal is define by ⎡ ⎤ Δf s ( t ) = Ac cos ⎢ 2πf c t + sin ( 2πf mt ) ⎥ fm ⎣ ⎦ ◊ Therefore, we may write ◊ Δf 2πk f ∫ m (τ ) dτ = ( sin 2πf mt ) 0 fm Differentiating both sides with respect to time and solving for m(t) t m (t ) = Δf cos ( 2πf mt ) kf 38 Yu-sing Lin (林育星) 6.5 Noise in FM Receivers ◊ ◊ ◊ ◊ The average power of message signal m(t) (Δf ) 2 P= 2k 2f We get output signal-to-noise ratio 2 3 Ac2 ( Δf ) 3 Ac2β 2 = (SNR )O ,FM = 3 4 N 0W 4 N 0W Where β = ∆f /W is the modulation index and we get the figure of 2 merit i (SNR )O 3 ⎛ Δf ⎞ 3 2 = ⎜ ( 6.43) ⎟ = β (SNR )C FM 2 ⎝ W ⎠ 2 It is important to note that the modulation index β = ∆f /W is determined de e ed by thee bandwidth ba dw d W oof thee postdetection pos de ec o low-pass ow pass filter and is related to the sinusoidal message frequency fm. 39 Yu-sing Lin (林育星) 6.5 Noise in FM Receivers ◊ For an AM system operating with a sinusoidal modulating signal and 100 percent modulation, we have (SNR )O 1 = (SNR )C AM 3 ◊ It is of particular interest to compare the noise performance of AM and FM systems. (SNR )O (SNR )C 3 ⎛ Δf = ⎜ 2⎝ W FM 3 2 1 when β > 2 3 ◊ 2 3 2 ⎞ = β ⎟ 2 ⎠ ←⎯⎯⎯ → compare (SNR )O (SNR )C = AM 1 3 FM has better performance 2 β> = 0.471 3 Define β = 0.5 0 5 as the transition between bet een narro narrowband band FM and wide-band FM. 40 Yu-sing Lin (林育星)