Lecture 1 6 Modulation 1 Bandpass Signalling Definitions

Lecture 1. 6. Modulation 1

Bandpass Signalling Ø Definitions Ø Complex Envelope Representation Ø Representation of Modulated Signals Ø Spectrum of Bandpass Signals Ø Power of Bandpass Signals Ø Examples

Bandpass Signals Ø Energy spectrum of a bandpass signal is concentrated around the carrier frequency fc. Bandpass Signal Spectrum Ø A time portion of a bandpass signal. Notice the carrier and the baseband envelope. Time Waveform of Bandpass Signal

DEFINITIONS The Bandpass communication signal is obtained by modulating a baseband analog or digital signal onto a carrier. Definitions: Ø A baseband waveform has a spectral magnitude that is nonzero for frequencies in the vicinity of the origin ( f=0) and negligible elsewhere. Ø A bandpass waveform has a spectral magnitude that is nonzero for frequencies in some band concentrated about a frequency where fc>>0. fc-Carrier frequency Ø Modulation is process of imparting the source information onto a bandpass signal with a carrier frequency fc by the introduction of amplitude or phase perturbations or both. Ø This bandpass signal is called the modulated signal s(t), and the baseband source signal is called the modulating signal m(t). Information Signal input processing m Carrier circuits Transmission medium (channel) Communication System Carrier circuits Signal processing

Complex Envelope Representation Ø The waveforms g(t) , x(t), R(t), and are all baseband waveforms. Additionally all of them except g(t) are real and g(t) is the Complex Envelope. • g(t) is the Complex Envelope of v(t) • x(t) is said to be the In-phase modulation associated with v(t) • y(t) is said to be the Quadrature modulation associated with v(t) • R(t) is said to be the Amplitude modulation (AM) on v(t) • (t) is said to be the Phase modulation (PM) on v(t) In communications, frequencies in the baseband signal g(t) are said to be heterodyned up to fc Ø THEOREM: Any physical bandpass waveform v(t) can be represented as below where fc is the CARRIER frequency and c=2 fc

Generalized transmitter using the AM–PM generation technique.

Generalized transmitter using the quadrature generation technique.

Complex Envelope Representation Ø THEOREM: Any physical bandpass waveform v(t) can be represented by where fc is the CARRIER frequency and c=2 fc PROOF: Any physical waveform may be represented by the Complex Fourier Series The physical waveform is real, and using , Thus we have: cn - negligible magnitudes for n in the vicinity of 0 and, in particular, c 0=0 Introducing an arbitrary parameter fc , we get v(t) – bandpass waveform with non-zero spectrum concentrated near f=fc => cn – non-zero for ‘n’ in the range => g(t) – has a spectrum concentrated near f=0 (i. e. , g(t) - baseband waveform)

Complex Envelope Representation Ø Equivalent representations of the Bandpass signals: Ø Converting from one form to the other form Inphase and Quadrature (IQ) Components. Envelope and Phase Components

Complex Envelope Representation Ø The complex envelope resulting from x(t) being a computer generated voice signal and y(t) being a sinusoid. The spectrum of the bandpass signal generated from above signal.

Representation of Modulated Signals Ø Modulation is the process of encoding the source information m(t) into a bandpass signal s(t). Modulated signal is just a special application of the bandpass representation. The modulated signal is given by: • The complex envelope g(t) is a function of the modulating signal m(t) and is given by: g(t)=g[m(t)] where g[ • ] performs a mapping operation on m(t). • The g[m] functions that are easy to implement and that will give desirable spectral properties for different modulations are given by the TABLE 4. 1 • At receiver the inverse function m[g] will be implemented to recover the message. Mapping should suppress as much noise as possible during the recovery.

Bandpass Signal Conversion • On off Keying (Amplitude Modulation) of a unipolar line coded signal for bandpass conversion. 1 Xn 0 Unipolar Line Coder cos( ct) 1 g(t) 0 X 1

Bandpass Signal Conversion • Binary Phase Shift keying (Phase Modulation) of a polar line code for bandpass conversion. 1 Xn 0 Polar Line Coder cos( ct) 1 g(t) 0 X 1

Mapping Functions for Various Modulations

Envelope and Phase for Various Modulations Eeng 360 15

Spectrum of Bandpass Signals Theorem: If bandpass waveform is represented by Where is PSD of g(t) Proof: Thus, Using We get, and the frequency translation property:

PSD of Bandpass Signals Ø PSD is obtained by first evaluating the autocorrelation for v(t): Using the identity where and We get - Linear operators => or but AC reduces to PSD =>

Evaluation of Power Theorem: Total average normalized power of a bandpass waveform v(t) is Proof: But So, or But So, is always real

Example : Amplitude-Modulated Signal Ø Evaluate the magnitude spectrum for an AM signal: Complex envelope of an AM signal: Spectrum of the complex envelope: AM signal waveform: AM spectrum: Magnitude spectrum:

Example : Amplitude-Modulated Signal Eeng 360 20 Spectrum of AM signal.

Example : Amplitude-Modulated Signal ØTotal average power:

Study Examples SA 4 -1. Voltage spectrum of an AM signal Properties of the AM signal are: g(t)=Ac[1+m(t)]; Ac=500 V; m(t)=0. 8 sin(2 1000 t); fc=1150 k. Hz; Fourier transform of m(t): Spectrum of AM signal: Substituting the values of Ac and M(f), we have

Study Examples SA 4 -2. PSD for an AM signal Autocorrelation for a sinusoidal signal (A sin w 0 t ) Autocorrelation for the complex envelope of the AM signal is Thus Using PSD for an AM signal:

Study Examples SA 4 -3. Average power for an AM signal Normalized average power Alternate method: area under PDF for s(t) Actual average power dissipated in the 50 ohm load: SA 4 -4. PEP for an AM signal Normalized PEP: Actual PEP for this AM voltage signal with a 50 ohm load:

Ø Ø Ø Bandpass Signalling Bandpass Filtering and Linear Distortion Bandpass Sampling Theorem Bandpass Dimensionality Theorem Amplifiers and Nonlinear Distortion Total Harmonic Distortion (THD) Intermodulation Distortion (IMD)

Bandpass Filtering and Linear Distortion Ø Equivalent Low-pass filter: Modeling a bandpass filter by using an equivalent low pass filter (complex impulse response) Bandpass filter Input bandpass waveform Output bandpass waveform Impulse response of the bandpass filter Frequency response of the bandpass filter H(f) = Y(f)/X(f)

Bandpass Filtering

Bandpass Filtering Theorem: The complex envelopes for the input, output, and impulse response of a bandpass filter are related by g 1(t) – complex envelope of input k(t) – complex envelope of impulse response Also, Proof: Spectrum of the output is Spectra of bandpass waveforms are related to that of their complex enveloped But

Bandpass Filtering Thus, we see that Taking inverse fourier transform on both sides Ø Any bandpass filter may be described analyzed by using an equivalent low-pass filter. Ø Equations for equivalent LPF are usually much less complicated than those for bandpass filters & so the equivalent LPF system model is very useful.

Linear Distortion For distortionless transmission of bandpass signals, the channel transfer function H(f) should satisfy the following requirements: Ø The amplitude response is constant A- positive constant Ø The derivative of the phase response is constant Tg – complex envelope delay Integrating the above equation, we get Are these requirements sufficient for distortionless transmission?

Linear Distortion

Linear Distortion The channel transfer function is If the input to the bandpass channel is Then the output to the channel (considering the delay Tg due to ) is Using Modulation on the carrier is delayed by Tg & carrier by Td Bandpass filter delays input info by Tg , whereas the carrier by Td

Bandpass Sampling Theorem: If a waveform has a non-zero spectrum only over the interval , where the transmission bandwidth BT is taken to be same as absolute BW, BT=f 2 -f 1, then the waveform may be reproduced by its sample values if the sampling rate is Quadrature bandpass representation Let fc be center of the bandpass: x(t) and y(t) are absolutely bandlimited to B=BT/2 The sampling rate required to represent the baseband signal is Quadrature bandpass representation now becomes Where and samples are independent , two sample values are obtained for each value of n Overall sampling rate for v(t):

Bandpass Dimensionality Theorem Ø Assume that a bandpass waveform has a nonzero spectrum only over a frequency interval , where the transmission bandwidth BT is taken to be the absolute bandwidth given by BT=f 2 -f 1 and BT<<f 1. ØThe waveform may be completely specified over a T 0 -second interval by N Independent pieces of information. N is said to be the number of dimensions required to specify the information.

Received Signal Pulse Information Signal input processing m Carrier circuits Transmission medium (Channel) Carrier circuits Signal processing The signal out of the transmitter g(t) – Complex envelope of v(t) If the channel is LTI , then received signal + noise n(t) – Noise at the receiver input Signal + noise at the receiver input A – gain of the channel - carrier phase shift caused by the channel, Tg – channel group delay. Signal + noise at the receiver input

Nonlinear Distortion Amplifiers Non-linear Linear Circuits with memory and circuits with no memory Memory - Present output value ~ function of present input + previous input values - contain L & C No memory - Present output values ~ function only of its present input values. Circuits : linear + no memory – resistive ciruits - linear + memory – RLC ciruits (Transfer function)

Nonlinear Distortion Assume no memory Present output as a function of present input in ‘t’ domain • If the amplifier is linear K- voltage gain of the amplifier • In practice, amplifier output becomes saturated as the amplitude of the input signal is increased. output-to-input characteristic (Taylor’s expansion): Where - output dc offset level - 1 st order (linear) term - 2 nd order (square law) term

Nonlinear Distortion Harmonic Distortion associated with the amplifier output: Let the input test tone be represented by To the amplifier input Then the second-order output term is = 2 nd Harmonic Distortion with In general, for a single-tone input, the output will be Vn – peak value of the output at the frequency nf 0 The Percentage Total Harmonic Distortion (THD) of an amplifier is defined by

Nonlinear Distortion Intermodulation distortion (IMD) of the amplifier: If the input (tone) signals are Then the second-order output term is Harmonic distortion at 2 f 1 & 2 f 2 Second-order IMD is: IMD

Nonlinear Distortion Third order term is The second term (cross-product) is The third term is Intermodulation terms at nonharmonic frequencies For bandpass amplifiers, where f 1 & f 2 are within the pasband, f 1 close to f 2, the distortion products at 2 f 1+f 2 and 2 f 2+f 1 ~ outside the passband Main Distortion Products

Bandpass Circuits Ø Limiters Ø Mixers, Upconverters and Downconverters Ø Detectors, Envelope Detector, Product Detector Ø Phase Locked Loops (PLL)

Limiters Ø Limiter is a nonlinear circuit with an output saturation characteristic. Ø It rejects envelope variations but preserves the phase variations. Ideal limiter characteristic with illustrative input and unfiltered output waveforms.

Mixers Ø Ideal mixer is a mathematical multiplier of two input signals. One of the signals is sinusoidal generated by a local oscillator. Mixing results in frequency translation. SSB mixer

Mixers

Choosing LO Frequency of Mixers Up-conversion Down-conversion Bandpass Filter Baseband/bandpass Filter (fc-f 0) Ø If (fc- f 0) = 0 Low Pass Filter gives baseband spectrum Ø If (fc- f 0 )> 0 Bandpass filter Modulation is preserved Filter Output: Ø If fc>f 0 modulation on the mixer input is preserved Ø If fc<f 0 Complex envelope is conjugated ~ sidebands are exchanged ‘ ’ needs to be positive

Mixers (Up Converter and Down Converter) Ø Complex envelope of an Up Converter: - Amplitude is scaled by A 0/2 Ø Complex envelope of a Down Converter: i. e. , f 0<fc down conversion with low-side injection - Amplitude is scaled by A 0/2 i. e. , f 0>fc down conversion with high-side injection - Amplitude is scaled by A 0/2 - Sidebands are reversed from those on the input

Mixer Realizations Without Multipliers Ø Multiplication operation needed by mixers can be obtained by using a nonlinear device together with a summer. Nonlinear device used as a mixer.

Mixer Realizations Without Multipliers Ø Multiplication operation needed by mixers can also be obtained by using an analog switch. Linear time-varying device used as a mixer.

Mixer Realizations Without Multipliers Analysis of a double-balanced mixer circuit.

Frequency Multiplier ØFrequency Multipliers consists of a nonlinear device together with a tuned circuit. The frequency of the output is n times the frequency of the input. Eeng 360 50

Detector Circuits Ø Detectors convert input bandpass waveform into an output baseband waveform. Ø Detector circuits can be designed to produce R(t), Θ(t), x(t) or y(t). • Envelope Detector • Product Detector • Frequency Modulation Detector Information Signal input processing m Carrier circuits Transmission medium (Channel) Detector Circuits Carrier circuits Signal processing

Envelope Detector Ø Ideal envelope detector: Waveform at the output is a real envelope R(t) of its input Bandpass input: Envelope Detector Output: K – Proportionality Constant Diode Envelope Detector Circuit

Envelope Detector Ø The Time Constant RC must be chosen so that the envelope variations can be followed. In AM, detected DC is used for Automatic Gain Control (AGC)

Product Detector Ø Product Detector is a Mixer circuit that down converts input to baseband. fc- Freq. of the oscillator θ 0 - Phase of the oscillator Output of the multiplier: LPF passes down conversion component: Where g(t) is the complex envelope of the input and x(t) & y(t) are the quadrature components of the input:

Different Detectors Obtained from Product Detector Ø Oscillator phase synchronized with the in-phase component We obtain INPHASE DETECTOR. Ø We obtain QUADRATURE PHASE DETECTOR Ø We obtain ENVELOPE DETECTOR If the input has no angle modulation and reference phase (θ 0) =0 Ø We obtain PHASE DETECTOR If an angle modulated signal is present at the input and reference phase (θ 0) =90 The product detector output is or If the phase difference is small The output is proportional to the Phase difference (Sinusoidal phase characteristics)

Frequency Modulation Detector Ø A ideal FM Detector is a device that produces an output that is proportional to the instantenous frequency of the input. Frequency demodulation using slope detection. • The DC output can easily be blocked

Frequency Detector Using Freq. to Amplitude Conversion Figure 4– 16 Slope detection using a single-tuned circuit for frequency-to amplitude conversion.

Balanced Discriminator

Balanced zero-crossing FM detector.

Phase Locked Loop (PLL) Ø PLL can be used to Track Phase and Frequency of the carrier component of the incoming signal Ø Three basic components: - Phase Detector : Multiplier (phase comparator) - VCO : Voltage Controlled Oscillator - Loop filter: LPF Ø Operation is similar to a feedback system Basic PLL.

PLL, Voltage Controlled Oscillator (VCO): Ø Oscillator frequency is controlled by external voltage Ø Oscillation frequency varies linearly with input voltage Ø If e 0(t) – VCO input voltage, then its output is a sinusoid of frequency (t)= c+ce 0(t) Ø c - free-running frequency of the VCO. Ø The multiplier output is further low-pass-filtered & then input to VCO Ø This voltage changes the frequency of the oscillator & keeps it locked.

Phase Locked Loop (PLL) Let input signal be : Let the VCO output be: The phase detector output v 1(t) is given by : The sum frequency term is rejected by LPF so the filter output v 2(t) is: Ø e(t) is called the Phase Error. The Phase Error voltage characteristics is SINUSOIDAL. ØA PLL can track the incoming frequency only over a finite range Lock/hold-in range Ø The frequency range over which the input will cause the loop to lock pull-in/capture range

Phase Locked Loop (PLL) Ø Various types of Phase Detector characteristics used in PLL’s.

Aplications of PLL Ø PLL used for coherent detection of AM signals. • A synchronized carrier signal is generated by the PLL. • VCO locks with 90 phase difference so a -90 extra phase shift is needed • The generated carrier is used with a product detector to recover the envelope Figure 4– 24 PLL used for coherent detection of AM.

Aplications of PLL Ø PLL used as a frequency synthesizer. Frequency dividers use integer values of M and N. For M=1 frequency synthesizer acts as a frequency multiplier. Figure 4– 25 PLL used in a frequency synthesizer.

Transmitters and Receivers Ø Generalized Transmitters Ø AM PM Generation Ø Inphase and Quadrature Generation Ø Superheterodyne Receiver Ø Frequency Division Multiplexing

Generalized Transmitters Modulating signal Transmitter Modulated signal Any type of modulated signal can be represented by The complex envelope g(t) is a function of the modulating signal m(t) Example:

Generalized Transmitters ØTwo canonical forms for the generalized transmitter: 1. AM- PM Generation Technique: Envelope and phase functions are generated to modulate the carrier as Figure 4– 27 Generalized transmitter using the AM–PM generation technique. R(t) and θ(t) are functions of the modulating signal m(t) as given in TABLE 4. 1

Generalized Transmitters 2. Quadrature Generation Technique: Inphase and quadrature signals are generated to modulate the carrier as Fig. 4– 28 Generalized transmitter using the quadrature generation technique. x(t) and y(t) are functions of the modulating signal m(t) as given in TABLE 4. 1

IQ (In-phase and Quadrature-phase) Detector

Generalized Receivers Ø Two types of receivers: Receivers Tuned Radio Frequency (TRF) Receiver: Composed of RF amplifiers and detectors. No frequency conversion It is not often used. Difficult to design tunable RF stages. Difficult to obtain high gain RF amplifiers Superheterodyne Receiver: Downconvert RF signal to lower IF frequency Main amplifixcation takes place at IF

Tuned Radio Frequency (TRF) Receivers Ø Composed of RF amplifiers and detectors. Ø No frequency conversion. It is not often used. Ø Difficult to design tunable RF stages. Ø Difficult to obtain high gain RF amplifiers Active Tuning Circuit Detector Circuit Local Oscillator Bandpass Filter Baseband Audio Amp

Heterodyning All Incoming Frequencies Heterodyning (Upconversion/ Downconversion) Fixed Intermediate Frequency Subsequent Processing (common)

Superheterodyne Receivers Superheterodyne Receiver Diagram

Superheterodyne Receiver

Superheterodyne Receivers Ø The RF and IF frequency responses H 1(f) and H 2(f) are important in providing the required reception characteristics.

Superheterodyne Receivers RF Response f. IF IF Response

Superheterodyne Receivers

Superheterodyne Receiver Frequencies

Superheterodyne Receiver Frequencies

Frequency Conversion Process

Image Frequencies Image frequency is also received Image frequency not a problem.

AM Radio Receiver

Superheterodyne Receiver Typical Signal Levels

Double-conversion block diagram.

Transmitter and Receiver Applications Ø AM and FM Radios Ø GSM and CDMA Ø TV Transmitters and Receivers Ø Direct Broadcast Satellite (DBS) TV

Complete 88 - to 108 -MHz stereo receiver. Eeng 360 87

Complete 88 - to 108 -MHz stereo receiver. Eeng 360 88

The GSM and CDMA networks. MSC = mobile switching center; BST = base transceiver station; BSC = base station controller.

MTSO linking two mobile users.

GSM Mobile Phone Transmitter Receiver

CDMA Cell Phone Mobile Phone Transmitter

TV receiver block diagram.

VHF/UHF tuner block diagram.

Video section block diagram.

Composite color TV transmission.

Direct Broadcast Satellite (DBS) TV

Direct Broadcast Satellite (DBS) TV

DBS-TV footprint and rain regions - 4 d. B - 6 d. B - 4 d. B - 2 d. B 150 o 120 o 90 o 60 o 30 o

DBS TV Uplink Transmitter QPSK modulator IF amplifier upconverter LPA HPA Multiplexer M ~ MUX ~ 70 MHz LO MPEG-2 encoder ADC Analog video and audio signals MPEG 2 Digital multiplexer MUX Other digital signals RF LO Reed-Soloman encoder R-S Coder Other RF signals 17 GHz uplink antenna Convolutional encoder Interleaver I Conv Coder

DBS TV Receiver 12. 2 – 12. 7 GHz DBS-TV signal LNA Image rejection BPF 900 - 1400 MHz IF amplifier Mixer Low noise block converter mounted on antenna feed Coaxial cable to set top receiver Local oscillator 11. 3 GHz Ku-band antenna 900 - 1400 MHz IF amplifier 70 MHz IF amplifier Mixer QPSK demod Baseband amplifier D Tuned BPF Select polarization transponder Inner decoder Tuned LO DI Deinterleaver Outer decoder input Frequency synthesizer Demux MPEG-2 decoder Microprocessor D/A Video Audio

DBS TV Receiver Front End Image rejection Mixer BPF Ku-band LNA antenna LO 11. 3 GHz LNB mounted on antenna feed Select polarization 900 - 1400 MHz IF amplifier Coaxial cable to set top receiver

DBS TV Receiver IF Section Coaxial cable from antenna and LNB Baseband 900 - 1400 MHz 70 MHz amplifier IF amplifier QPSK Mixer demod D in Tuned BPF F P to select transponder Tuned LO Frequency Microprocessor synthesizer

DBS TV Receiver Baseband Processing Deinterleaver bits CD DI Inner convolutional decoder R-SD Outer R-S decoder De. Mux MPEG Digital demux MPEG-2 decoder D/A Video Audio Analog outputs

END