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Page 1 Modulation and Demodulation of Analog Signals Martin Kumm March 5, 2009 Contents 1 Introduction 2 2 Mathematical Representation of Modulated Signals 2 3 Modulation 3 3.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Single Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Phase and Frequency Modulation . . . . . . . . . . . . . . . . . . . . 5 4 Demodulation 5 4.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 Single Sideband Demodulation . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Phase and Frequency Demodulation . . . . . . . . . . . . . . . . . . . 11 References 11 1 Page 2 Modulation and Demodulation of Analog Signals Martin Kumm March 5, 2009 Contents 1 Introduction 2 2 Mathematical Representation of Modulated Signals 2 3 Modulation 3 3.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Single Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Phase and Frequency Modulation . . . . . . . . . . . . . . . . . . . . 5 4 Demodulation 5 4.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 Single Sideband Demodulation . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Phase and Frequency Demodulation . . . . . . . . . . . . . . . . . . . 11 References 11 1 1 Introduction The motivation for writing this article was that I started implementing blocks for demodulating RF signals on Field Programmable Gate Arrays (FPGAs). During that I was asking me what are the best methods and structures for doing that. Most of the text books dealing with the topic of receivers either focussing on practical concerns of analog receivers or have a mathematical point of view that is hard to realize in hardware. So I started some calculations that are summarized in this document. 2 Mathematical Representation of Modulated Signals An unmodulated radio frequency (RF) signal can be described as cosine waveform oscillation at an RF frequency f RF = ! RF =(2) with a peak amplitude ^ a and an initial phase 0 1 : s RF (t) = ^ a cos(! RF t + 0 ) (1) This RF waveform can be independently modulated in amplitude and phase (the argument of the cosine), leading to the representation s RF (t) = ^ a(t) cos ['(t)] : (2) A further representation that is very usefull can be found by extending the real signal with an imaginary part which is its own Hilbert Transform (e.g. see [Kam04],[KK06],[OS99]) leading to the so-called analytic signal s RF (t) =s RF (t) +jHfs RF (t)g : (3) In the case of a cosine function, the Hilbert Transform is a sine function s RF (t) = ^ a(t) cos ['(t)] +j sin ['(t)] = ^ a(t)e j'(t) : (4) The analytic signal is marked with an underline in the following. The real RF wave- form can be obtained by simply taking the real part of the analytic signal s RF (t) =Refs RF (t)g : (5) The amplitude and phase components for common analog modulation modes are summarized in Table 1. The low frequency modulation signal is called s LF and is normalized to an absolute peak value of one. The signals s + LF (t) and s LF (t) are the upper and lower sidebands of the analytic representation of the modulation signal. The upper one is the natural representation of an analytic signal s + LF (t) =s LF (t) =s LF (t) +jHfs LF (t)g (6) 1 A representation using the sine function can be made by simply changing the denition of 0 . Using the cosine function eases the converting between real an complex signals. 2 Page 3 Modulation and Demodulation of Analog Signals Martin Kumm March 5, 2009 Contents 1 Introduction 2 2 Mathematical Representation of Modulated Signals 2 3 Modulation 3 3.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Single Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Phase and Frequency Modulation . . . . . . . . . . . . . . . . . . . . 5 4 Demodulation 5 4.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 Single Sideband Demodulation . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Phase and Frequency Demodulation . . . . . . . . . . . . . . . . . . . 11 References 11 1 1 Introduction The motivation for writing this article was that I started implementing blocks for demodulating RF signals on Field Programmable Gate Arrays (FPGAs). During that I was asking me what are the best methods and structures for doing that. Most of the text books dealing with the topic of receivers either focussing on practical concerns of analog receivers or have a mathematical point of view that is hard to realize in hardware. So I started some calculations that are summarized in this document. 2 Mathematical Representation of Modulated Signals An unmodulated radio frequency (RF) signal can be described as cosine waveform oscillation at an RF frequency f RF = ! RF =(2) with a peak amplitude ^ a and an initial phase 0 1 : s RF (t) = ^ a cos(! RF t + 0 ) (1) This RF waveform can be independently modulated in amplitude and phase (the argument of the cosine), leading to the representation s RF (t) = ^ a(t) cos ['(t)] : (2) A further representation that is very usefull can be found by extending the real signal with an imaginary part which is its own Hilbert Transform (e.g. see [Kam04],[KK06],[OS99]) leading to the so-called analytic signal s RF (t) =s RF (t) +jHfs RF (t)g : (3) In the case of a cosine function, the Hilbert Transform is a sine function s RF (t) = ^ a(t) cos ['(t)] +j sin ['(t)] = ^ a(t)e j'(t) : (4) The analytic signal is marked with an underline in the following. The real RF wave- form can be obtained by simply taking the real part of the analytic signal s RF (t) =Refs RF (t)g : (5) The amplitude and phase components for common analog modulation modes are summarized in Table 1. The low frequency modulation signal is called s LF and is normalized to an absolute peak value of one. The signals s + LF (t) and s LF (t) are the upper and lower sidebands of the analytic representation of the modulation signal. The upper one is the natural representation of an analytic signal s + LF (t) =s LF (t) =s LF (t) +jHfs LF (t)g (6) 1 A representation using the sine function can be made by simply changing the denition of 0 . Using the cosine function eases the converting between real an complex signals. 2 as it has only signal components for positive frequencies. The lower sideband is the complex conjugate of the analytic signal: s LF (t) =s LF (t) =s LF (t)jHfs LF (t)g (7) Furthermore, the modulation modes are characterized by the parameters a 0 , which is the peak RF amplitude, the modulation index m (m =f0:::1g), the peak phase variation and the peak frequency variation !. It should be noticed that the AM mode only in uences the amplitude and PM and FM only in uences the phase while the single sideband modulation modes USB and LSB in uences both magnitudes [Kam04]. 3 Modulation In this section, the modulation and demodulation schemes are shown for continuous signals and ideal components (e.g. lters, oscillators, etc.). 3.1 Amplitude Modulation The structure of an amplitude modulator is shown in Figure 1. It is the direct realization of the formula for amplitude modulation in Table 1. The corresponding spectra of intermediate signals are shown in Figure 2. For a better understanding, the intermediate signals of Figure 1 are calculated for a modulating sinwave signal s LF (t) = cos(! LF t + LF ) : (8) s RF (t) = cos(! RF t) 1 2 + m 2 cos(! LF t + LF ) = 1 2 cos(! RF t) + m 4 cos((! RF +! LF )t + LF ) + m 4 cos((! RF ! LF )t LF ) (9) Modulation Mode ^ a(t) '(t) Amplitude Modulation (AM) a 0 1 2 [1 +ms LF (t)] ! RF t + 0 Upper Sideband Modulation (USB) a 0 m s + LF (t) ! RF t + 0 Lower Sideband Modulation (LSB) a 0 m s LF (t) ! RF t + 0 Phase Modulation (PM) a 0 ! RF t + s LF (t) + 0 Frequency Modulation (FM) a 0 [! RF !s LF (t)]t + 0 Table 1: Modulation Modes 3 Page 4 Modulation and Demodulation of Analog Signals Martin Kumm March 5, 2009 Contents 1 Introduction 2 2 Mathematical Representation of Modulated Signals 2 3 Modulation 3 3.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Single Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Phase and Frequency Modulation . . . . . . . . . . . . . . . . . . . . 5 4 Demodulation 5 4.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 Single Sideband Demodulation . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Phase and Frequency Demodulation . . . . . . . . . . . . . . . . . . . 11 References 11 1 1 Introduction The motivation for writing this article was that I started implementing blocks for demodulating RF signals on Field Programmable Gate Arrays (FPGAs). During that I was asking me what are the best methods and structures for doing that. Most of the text books dealing with the topic of receivers either focussing on practical concerns of analog receivers or have a mathematical point of view that is hard to realize in hardware. So I started some calculations that are summarized in this document. 2 Mathematical Representation of Modulated Signals An unmodulated radio frequency (RF) signal can be described as cosine waveform oscillation at an RF frequency f RF = ! RF =(2) with a peak amplitude ^ a and an initial phase 0 1 : s RF (t) = ^ a cos(! RF t + 0 ) (1) This RF waveform can be independently modulated in amplitude and phase (the argument of the cosine), leading to the representation s RF (t) = ^ a(t) cos ['(t)] : (2) A further representation that is very usefull can be found by extending the real signal with an imaginary part which is its own Hilbert Transform (e.g. see [Kam04],[KK06],[OS99]) leading to the so-called analytic signal s RF (t) =s RF (t) +jHfs RF (t)g : (3) In the case of a cosine function, the Hilbert Transform is a sine function s RF (t) = ^ a(t) cos ['(t)] +j sin ['(t)] = ^ a(t)e j'(t) : (4) The analytic signal is marked with an underline in the following. The real RF wave- form can be obtained by simply taking the real part of the analytic signal s RF (t) =Refs RF (t)g : (5) The amplitude and phase components for common analog modulation modes are summarized in Table 1. The low frequency modulation signal is called s LF and is normalized to an absolute peak value of one. The signals s + LF (t) and s LF (t) are the upper and lower sidebands of the analytic representation of the modulation signal. The upper one is the natural representation of an analytic signal s + LF (t) =s LF (t) =s LF (t) +jHfs LF (t)g (6) 1 A representation using the sine function can be made by simply changing the denition of 0 . Using the cosine function eases the converting between real an complex signals. 2 as it has only signal components for positive frequencies. The lower sideband is the complex conjugate of the analytic signal: s LF (t) =s LF (t) =s LF (t)jHfs LF (t)g (7) Furthermore, the modulation modes are characterized by the parameters a 0 , which is the peak RF amplitude, the modulation index m (m =f0:::1g), the peak phase variation and the peak frequency variation !. It should be noticed that the AM mode only in uences the amplitude and PM and FM only in uences the phase while the single sideband modulation modes USB and LSB in uences both magnitudes [Kam04]. 3 Modulation In this section, the modulation and demodulation schemes are shown for continuous signals and ideal components (e.g. lters, oscillators, etc.). 3.1 Amplitude Modulation The structure of an amplitude modulator is shown in Figure 1. It is the direct realization of the formula for amplitude modulation in Table 1. The corresponding spectra of intermediate signals are shown in Figure 2. For a better understanding, the intermediate signals of Figure 1 are calculated for a modulating sinwave signal s LF (t) = cos(! LF t + LF ) : (8) s RF (t) = cos(! RF t) 1 2 + m 2 cos(! LF t + LF ) = 1 2 cos(! RF t) + m 4 cos((! RF +! LF )t + LF ) + m 4 cos((! RF ! LF )t LF ) (9) Modulation Mode ^ a(t) '(t) Amplitude Modulation (AM) a 0 1 2 [1 +ms LF (t)] ! RF t + 0 Upper Sideband Modulation (USB) a 0 m s + LF (t) ! RF t + 0 Lower Sideband Modulation (LSB) a 0 m s LF (t) ! RF t + 0 Phase Modulation (PM) a 0 ! RF t + s LF (t) + 0 Frequency Modulation (FM) a 0 [! RF !s LF (t)]t + 0 Table 1: Modulation Modes 3 Figure 1: Amplitude modulation structure Figure 2: Spectra of intermediate signals for amplitude modulation 3.2 Single Sideband Modulation As Table 1 shows, the single sideband modulation can be realized by taking the Hilbert Transform of the modulation signal followed by a complex multiplication with the complex RF generator signal. This structure is shown in Figure 3. The upper sideband (USB) can be selected by a plus sign and the lower sideband (LSB) by the minus sign. The spectra of the intermediate signals of Figure 3 are shown in Figure 4. The output is a complex signal { but we are only interested in the real part of that signal. This simplies the complex multiplication as the real part of the multiplication result Figure 3: SSB modulation structure 4 Page 5 Modulation and Demodulation of Analog Signals Martin Kumm March 5, 2009 Contents 1 Introduction 2 2 Mathematical Representation of Modulated Signals 2 3 Modulation 3 3.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Single Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Phase and Frequency Modulation . . . . . . . . . . . . . . . . . . . . 5 4 Demodulation 5 4.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 Single Sideband Demodulation . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Phase and Frequency Demodulation . . . . . . . . . . . . . . . . . . . 11 References 11 1 1 Introduction The motivation for writing this article was that I started implementing blocks for demodulating RF signals on Field Programmable Gate Arrays (FPGAs). During that I was asking me what are the best methods and structures for doing that. Most of the text books dealing with the topic of receivers either focussing on practical concerns of analog receivers or have a mathematical point of view that is hard to realize in hardware. So I started some calculations that are summarized in this document. 2 Mathematical Representation of Modulated Signals An unmodulated radio frequency (RF) signal can be described as cosine waveform oscillation at an RF frequency f RF = ! RF =(2) with a peak amplitude ^ a and an initial phase 0 1 : s RF (t) = ^ a cos(! RF t + 0 ) (1) This RF waveform can be independently modulated in amplitude and phase (the argument of the cosine), leading to the representation s RF (t) = ^ a(t) cos ['(t)] : (2) A further representation that is very usefull can be found by extending the real signal with an imaginary part which is its own Hilbert Transform (e.g. see [Kam04],[KK06],[OS99]) leading to the so-called analytic signal s RF (t) =s RF (t) +jHfs RF (t)g : (3) In the case of a cosine function, the Hilbert Transform is a sine function s RF (t) = ^ a(t) cos ['(t)] +j sin ['(t)] = ^ a(t)e j'(t) : (4) The analytic signal is marked with an underline in the following. The real RF wave- form can be obtained by simply taking the real part of the analytic signal s RF (t) =Refs RF (t)g : (5) The amplitude and phase components for common analog modulation modes are summarized in Table 1. The low frequency modulation signal is called s LF and is normalized to an absolute peak value of one. The signals s + LF (t) and s LF (t) are the upper and lower sidebands of the analytic representation of the modulation signal. The upper one is the natural representation of an analytic signal s + LF (t) =s LF (t) =s LF (t) +jHfs LF (t)g (6) 1 A representation using the sine function can be made by simply changing the denition of 0 . Using the cosine function eases the converting between real an complex signals. 2 as it has only signal components for positive frequencies. The lower sideband is the complex conjugate of the analytic signal: s LF (t) =s LF (t) =s LF (t)jHfs LF (t)g (7) Furthermore, the modulation modes are characterized by the parameters a 0 , which is the peak RF amplitude, the modulation index m (m =f0:::1g), the peak phase variation and the peak frequency variation !. It should be noticed that the AM mode only in uences the amplitude and PM and FM only in uences the phase while the single sideband modulation modes USB and LSB in uences both magnitudes [Kam04]. 3 Modulation In this section, the modulation and demodulation schemes are shown for continuous signals and ideal components (e.g. lters, oscillators, etc.). 3.1 Amplitude Modulation The structure of an amplitude modulator is shown in Figure 1. It is the direct realization of the formula for amplitude modulation in Table 1. The corresponding spectra of intermediate signals are shown in Figure 2. For a better understanding, the intermediate signals of Figure 1 are calculated for a modulating sinwave signal s LF (t) = cos(! LF t + LF ) : (8) s RF (t) = cos(! RF t) 1 2 + m 2 cos(! LF t + LF ) = 1 2 cos(! RF t) + m 4 cos((! RF +! LF )t + LF ) + m 4 cos((! RF ! LF )t LF ) (9) Modulation Mode ^ a(t) '(t) Amplitude Modulation (AM) a 0 1 2 [1 +ms LF (t)] ! RF t + 0 Upper Sideband Modulation (USB) a 0 m s + LF (t) ! RF t + 0 Lower Sideband Modulation (LSB) a 0 m s LF (t) ! RF t + 0 Phase Modulation (PM) a 0 ! RF t + s LF (t) + 0 Frequency Modulation (FM) a 0 [! RF !s LF (t)]t + 0 Table 1: Modulation Modes 3 Figure 1: Amplitude modulation structure Figure 2: Spectra of intermediate signals for amplitude modulation 3.2 Single Sideband Modulation As Table 1 shows, the single sideband modulation can be realized by taking the Hilbert Transform of the modulation signal followed by a complex multiplication with the complex RF generator signal. This structure is shown in Figure 3. The upper sideband (USB) can be selected by a plus sign and the lower sideband (LSB) by the minus sign. The spectra of the intermediate signals of Figure 3 are shown in Figure 4. The output is a complex signal { but we are only interested in the real part of that signal. This simplies the complex multiplication as the real part of the multiplication result Figure 3: SSB modulation structure 4 can be realized by two real multiplications and one real subtraction: Ref(a r +ja i ) (b r +jb i )g =Refa r b r a i b i +j(a i b r +a r b i )g =a r b r a i b i (10) This is shown in Figure 5. The upper sideband (USB) is now selected by a minus sign and the lower sideband (LSB) by the plus sign. For a better understanding, the intermediate signals of Figure 5 are calculated for a modulating sinwave signal s LF (t) = cos(! LF t + LF ) : (11) The Hilbert Transform of the input signal is Hfs LF (t)g = sin(! LF t + LF ) : (12) Multiplying the input signal with the cosine RF waveform results in s LF (t) cos(! RF t) = 1 2 cos((! RF +! LF )t + LF ) + 1 2 cos((! RF ! LF )t LF ) : (13) Multiplying the Hilbert Transform of the input signal with the sine RF waveform results in Hfs LF (t)g cos(! RF t) = 1 2 sin((! RF +! LF )t + LF ) + 1 2 sin((! RF ! LF )t LF ) : (14) Subtracting these signals results in the USB signal s + RF (t) = cos((! RF +! LF )t + LF ) ; (15) while an addition results in the LSB signal s RF (t) = cos((! RF ! LF )t LF ) : (16) 3.3 Phase and Frequency Modulation The phase and frequency modulation is a manipulation of the argument of the RF waveform and must therefore be realized directly in the oscillator i.e. during gener- ation of the RF signal. Due to the principle of direct digital synthesis (DDS) (for a detailed description about DDS see e.g. [Kro98]) this can be easily realized by adding the modulation signal to the phase or frequency word. 4 Demodulation 4.1 Amplitude Demodulation Simple Product Detector 5Read More

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