ECEN 242 Wireless Electronics for Communication Spring 22-3-2 P. Mathys Wireless Communication Brief History In 893 Nikola Tesla (Serbian-American, 856 943) gave lectures in Philadelphia before the Franklin Institute and in St. Louis before the National Electrical Light Association about wireless radio telegraphy and about light and other high frequency phenomena. His interest in high frequencies was twofold, wireless communication and wireless transmission of energy (e.g., for electrical lighting). The first entrepreneur to successfully commercialize wireless was Guglielmo Marconi (Italian, 874 937) who founded the Wireless Telegraph & Signal Company in 897. In 94 his company, renamed as Marconi Wireless Telegraph, established a commercial service to transmit news and telegrams to subscribing steamships. Initially, wireless communication was only used to exchange messages in Morse code between trained operators. The potential to broadcast voice and music signals was not recognized until after World War I in 98. The first regular AM radio broadcast station in the US was KDKA in Pittsburgh, PA. It opened on election night, November 2, 92, to broadcast the presidential election results. FM radio was patented in 933 by Edwin Howard Armstrong (American, 89 954) and he began regular programming in July 939. Regular television broadcasting started in North America and Europe in the 94s. 2 Major Modulation Systems A sinusoid of the form x(t) = A cos(2πft + θ), is characterized by its amplitude A, its frequency f, and its phase θ. If one or more of these quantities is varied in proportion to a message signal m(t), then x(t) can act as a carrier for the transmission of m(t). Amplitude modulation (AM) in the strict sense, when neither f nor θ are affected, is obtained by setting x(t) = A c ( + αm n (t)) cos(2πf c t + θ c ), where A c is the carrier amplitude, α is the modulation index, m n (t) is the normalized (i.e., m n (t) ) message signal, f c is the carrier frequency, and θ c is the carrier phase. An example of such an AM signal is shown below.
Amplitude Modulation (AM), A c =.5, f c =2 Hz, f m =2 Hz, α=.66667 Message Signal m n (t).5.5 2 3 4 5 6 7 8 9 AM Signal x(t).5.5 2 3 4 5 6 7 8 9 t [ms] Phase modulation (PM) with message signal m n (t) is obtained by using x(t) = A c cos ( 2πf c t + θ c + θ m n (t) ), where f c is the carrier frequency, θ c is the carrier phase, and θ is the maximum phase deviation. An example is shown in the following graph. 2
Phase Modulation (PM), f c =2 Hz, f m =2 Hz, θ =.7854 rad Message Signal m(t).5.5 2 3 4 5 6 7 8 9 PM Signal x(t).5.5 2 3 4 5 6 7 8 9 t [ms] Note the dotted green waveform in the lower graph which is the unmodulated carrier that is shown as reference so that the phase changes can be seen. The instantaneous frequency f i (t) in Hz of a sinusoid of the form A c cos ( 2πf c t + θ c + ψ(t) ) is defined as f i (t) = d [ 2πfc t + θ c + ψ(t) ] = f c + d [ ] ψ(t). 2π dt 2π dt Frequency modulation (FM) with a message signal m n (t) is obtained by varying the instantaneous frequency in proportion to m n (t) over a specified maximum range. This is achieved by setting d [ ] ψ(t) = f m n (t), dt where f is the maximum instantaneous frequency deviation in Hz. After integration and adding a carrier frequency f c and a carrier phase θ c, the FM signal then becomes x(t) = A c cos ( 2πf c t + θ c + 2π f t m n (τ) dτ ). Depending on the relationship between f and the highest message frequency f m, a distinction is made between narrowband ( f less than f m ) and wideband FM ( f greater than f m ). The graph below shows an example of a wideband FM signal. 3
Frequency Modulation (FM), f c =2 Hz, f m =2 Hz, f = Hz Message Signal m(t).5.5 2 3 4 5 6 7 8 9 FM Signal x(t).5.5 2 3 4 5 6 7 8 9 t [ms] Regular FM broadcast radio (87.5 to 8 MHz band in the US) uses a maximum frequency deviation of 75 khz and a maximum message frequency of 5 khz and is therefore classified as wideband FM. 3 Frequency and Wavelength The wavelength λ in meters of a signal with frequency f in Hertz is where c = 3 8 m/s is the speed of light. λ = c f = 3 8 f, 4 Amateur Radio 4
5 Complex Numbers and Euler s Formula Complex numbers arise from the question: What is the solution for x 2 + =? Clearly, x 2 = is needed, but what is? The answer is that does not exist in the set of real numbers and therefore the number system needs to be extended to include complex numbers of the form x = a + j b, where j =. Note that j 2 =. The quantity a is called the real part of x, denoted by Re{x}, and b is called the imaginary part of x, denoted by Im{x}. Note that j itself is the imaginary unit and is not part of Im{x}. The complex conjugate of x, denoted by x is x = a j b, if x = a + j b. The magnitude (or length) of a complex number is defined as x = x x = (a + j b)(a j b) = a 2 + b 2 = Re 2 {x} + Im 2 {x}. An important mathematical relationship that has many applications in engineering is Euler s relationship or formula e jθ = cos θ + j sin θ, where e = 2.78... is the base of the natural logarithm. Since cos π = and sin π = this also leads to Euler s identity e jπ + =, which in very compact form captures much of the essence of mathematics. Addition, multiplication, and exponentiation each occur once. The identity elements with respect to addition () and with respect to multiplication () are present, as well as the important mathematical constants e and π. And last but not least the imaginary unit j (or i outside of electrical engineering) which is important in both algebra and calculus. Since cos 2 θ + sin 2 θ =, e jθ =, i.e., for all real values of θ e jθ is a point on the unit circle. Thus, Euler s formula can be interpreted graphically as shown below. Im sin θ e jθ θ cos θ Re 5
More generally, a complex number x can be represented in polar form as or in cartesian form as x = r e jθ, where x = r, and x = θ, x = a + j b, where Re{x} = a, and Im{x} = b. The conversion between the two formats is a = r cos θ, b = r sin θ, and r = a 2 + b 2, θ = tan ( b ). a Two interchangeable notations that are frequently used are e jπ/2 = j, and e jπ/2 = j. Using either the graphical representation or the mathematical formulas, the validity of these expressions is easily checked. Another consequence of Euler s formula is the following complicated way of expressing e j2π =, or, more generally, e j2πk =, which holds because cos(2πk) = and sin(2πk) = for all integer values of k. This is useful to find all roots of the expression N ( N-th root of unity ) using N = () /N = (e j2πk ) /N = e j2πk/n, k =,,..., N. Graphically, the interpretation is that the N solutions of N all lie on the unit circle, spaced 2π/N radians apart. 6 Waveforms A signal or waveform is a function of an independent variable. For communications and signal processing, waveforms occur often as functions of time t. An example of a waveform is the sinusoidal signal x(t) = A cos(2πft + θ) that we have seen already. An important signal that is used for practical measurements as well as a test signal and building block for theoretical considerations is the unit step function u(t). It is defined as follows: u(t) u(t) = {, t >,, t <. t 6
Note that u(t) has a discontinuity at t = and the value of u() is not specified in general. A waveform x(t) can be shifted t > time units to the right by replacing x(t) with x(t t ). For the unit step this looks as follows: u(t t ) u(t t ) =, t > t,, t < t. t t Similarly, a waveform x(t) can be shifted t > time units to the left by replacing x(t) with x(t + t ). The result for the unit step is shown below. u(t + t ) = u(t + t ), t > t,, t < t. t t To generate a rectangular pulse p(t) of width T, two unit step functions can be combined as shown next. p(t) p(t) = u(t) u(t T ) =, <t<t,, otherwise. t T A waveform that does not exist physically but is of great mathematical importance is the unit impulse or delta function δ(t). It can be defined as the derivative of the unit step function: Z t du(t) = δ(τ ) dτ. δ(t) = dt A slightly different definition and a graphical representation for δ(t) are shown below. δ(t) () δ(t) =, if t =, and δ(τ ) dτ =, all > 7 t
The most important features of δ(t) are that it is whenever t and that it has area, hence the () for the size of δ(t) in the graph. The unit impulse is not a waveform in the conventional sense where x(t) is defined for every t. To approximate δ(t) for practical purposes (e.g., in computer simulations) it is possible to use a tall narrow pulse with area, e.g., a rectangular pulse of width ɛ and height /ɛ, where ɛ. Note that δ(t) can be shifted left by writing δ(t t ), and right by writing δ(t + t ). A useful property is the sifting property of the delta function which says that for x(t) which has no discontinuity at t = t x(t) δ(t t ) = x(t ) δ(t t ). Using conventional mathematics this equality is questionable (hence the quotation marks), but intuitively the result can be justified because δ(t t ) is zero for all t t. Therefore it picks out the value of x(t) at t = t. If the unit step is integrated, then unit ramp function r(t) is obtained: r(t) = t u(τ) dτ. The analytical characterization and the graph of r(t) are shown below. r(t) = { t, t,, t <. r(t).... t Again, it is possible to left and right shift the ramp by using r(t t ) and r(t+t ), respectively. This can be used to construct the triangular pulse p(t) of width 2T ahown below. p(t) =r(t) 2r(t T )+r(t 2T ) T p(t) T 2T t More complex waveforms can be generated by using combinations of scaled and shifted unit step and unit ramp functions. 7 Time and Frequency Domains An expression of the form x(t) = cos(2πf t + θ), 8
defines a waveform x(t) in the time domain for all values of time t. But we can also say that x(t) is a sinusoid with frequency f and phase θ. It could be used as a carrier for a radio signal in the AM radio band if f is in the range of 52 khz to.6 MHz. Many radio stations can broadcast simultaneously without interfering with each other in this band if they use different carrier frequencies. This approach is called frequency division multiplexing (FDM). Consequently, characterizations of signals based on their frequency and phase are called frequency domain characterizations. In 87 Joseph Fourier (768 83), while solving problems of heat transfer and vibration, claimed that any periodic function of a variable, whether continuous or discontinuous, can be represented as a weighted sum of much simpler sinusoidal component functions. Although this statement is not true for any periodic function that can be constructed mathematically, it turns out to be true for most physical functions of practical interest. The complex-valued version of the Fourier series, as the representation is now called in honor of Fourier, is defined as follows: Definition: The Fourier Series (FS) of a periodic continuous time signal x(t) with period T is defined as X k = T T x(t) e j2πkt/t dt, k =, ±, ±2,..., where the integration is taken over any interval of length T. The FS coefficients X k correspond to frequency components at f k = k/t. Frequency f = /T is called the fundamental frequency, f 2 = 2/T is called the 2 nd harmonic, f 3 = 3/T is called the 3 rd harmonic, etc. Theorem: Inverse FS. A periodic CT signal x(t) can be recovered uniquely from its FS coefficients X k (provided that they exist) by where T is the period of x(t). x(t) = k= X k e j2πkt/t, Example: Periodic rectangular waveform x(t) with amplitude, 5% duty cycle, and period T defined by {, mt T x(t) = /4 t < mt + T /4, m integer,, otherwise. The FS coefficients X k are computed as X k = T T x(t) e j2πkt/t dt = T T /4 To synthesize x(t) from X k the formula x(t) = T /4 e j2πkt/t dt = sin(πk/2) πk k= 9 X k e j2πkt/t,, k =, ±, ±2,....
is used. However, in practice only a limited number of X k may be available, e.g., in the range K max k K max for some finite integer K max. The two graphs below show x(t) and X k (magnitude and phase) for the rectangular waveform when K max = 5..2 Fourier Series Representation of Rectangular Pulse K max =5.8.6.4 x(t).2.2.4.6.8.8.6.4.2.2.4.6.8 t [ms]
.5.4 X k.3.2. 5 4 3 2 2 3 4 5 2 5 X k [deg] 5 5 4 3 2 2 3 4 5 k The next two graphs show x(t) and X k for the same waveform when K max is increased to 5..2 Fourier Series Representation of Rectangular Pulse K max =5.8.6.4 x(t).2.2.4.6.8.8.6.4.2.2.4.6.8 t [ms]
.5.4 X k.3.2. 5 4 3 2 2 3 4 5 2 5 X k [deg] 5 5 4 3 2 2 3 4 5 k 8 Block Diagrams Block diagrams are widely used in engineering to convey ideas and concepts and to specify functions and relationships of and among electrical and mechanical systems and subsystems. Block diagrams generally specify systems at a higher level of abstraction that often makes use of idealizations to simplify the exposition and make the description independent of a particular implementation. Usually, a system described by a block diagram can be implemented in may different ways, e.g., using analog or digital circuitry in electrical engineering. To specify the details of an actual implementation, schematic diagrams and detailed component descriptions are used. Here is an example of an actual circuit that computes OUT=9 IN- 2 IN2. 2
The corresponding blockdiagram, which computes y(t) = 9 x (t) 2 x 2 (t), is shown below. x (t) 9 x 2 (t) 2 + + y(t) Clearly, blockdiagrams have the capability to convey information at a higher level in a much more general and compact form. Some of the common symbols used in blockdiagrams are shown in the following table. 3
Symbol x(t) Function A y(t) Multiplication by a Constant y(t) = A x(t) x (t) x2 (t) x (t) x2 (t) + + + y(t) Addition y(t) = x (t) + x2 (t) + + y(t) Subtraction y(t) = x (t) x2 (t) y(t) Multiplication y(t) = x (t) x2 (t) x (t) x2 (t) x(t) y(t) Integration Rt y(t) = x(τ ) dτ x(t) x(t) x(t) LPF at fl y(t) BPF fc, W y(t) HPF at fh y(t) (Ideal) Lowpass Filter with Cutoff Frequency fl (Ideal) Bandpass Filter Center Frequency fc, Bandwidth W 4 Filter (Ideal) Highpass with Cutoff Frequency fh
9 Amplitude Modulation In the form of a block diagram, amplitude modulation (AM) of a carrier A c cos(2πf c t + θ c ), with carrier frequency f c and carrier phase θ c, with a signal s(t) can be characterized as follows. s(t) x(t) A c cos(2πf c t + θ c ) Analytically, x(t) = A c s(t) cos(2πf c t + θ c ). If s(t) for all t, then only the amplitude of the carrier is modified by s(t). To transmit Morse code signals, s(t) is either on (short for dots, long for dashes ) or off (in between dots, dashes, and letters made from dots and dashes). The resulting modulation is called CW (for continuous wave) or OOK (on/off keying). An example, using the Morse code for CQ ( seek you ) is shown in the figure below..5 CW or OOK (AM) Signal, A c =, f c =4 Hz, θ c = deg Signal s(t).5 5 5.5 CW Signal x(t).5.5.5 5 5 t [ms] 5
Commercial AM radio broadcasting in the 54 to 6 khz band uses s(t) = + α m n (t), where m n (t) = m(t) max τ ( m(τ) ) is the normalized message signal, m(t) is the speech or music signal to be transmitted, and α is the modulation index. Note that ( x(t) = A c + αmn (t) ) cos(2πf c t + θ c ) = A c cos(2πf c t + θ c ) + αa }{{} c m n (t) cos(2πf c t + θ c ) }{{} carrier term sidebands The carrier term does not depend on m n (t) and is therefore useless from the point of view of transmitting the message m(t).. Continuous-Time and Discrete-Time Signals A function or signal x(t) that is defined for all instants of time in some interval, such as the sinusoid x(t) = A cos(2πft + θ), < t <, is called a continuous-time (CT) function or signal. By sampling such an x(t) at time instants t = nt s, n =..., 2,,,, 2,..., a discrete-time (DT) function or signal, usually denoted by x n or x[n], is obtained for which the values are only known at integer multiples of the sampling interval T s. We set x n = x[n] = x(nt s ) and say that the signal x(t) has been sampled with sampling frequency F s = /T s in samples per second or Hertz. An example of a sine signal, sampled 8 times per period, is shown in the graph below. CT Signal x(t)=sin(2πf t) and DT Signal x n =x(nt s ), f = Hz, F s =/T s =8 Hz x(t), x n =x(nt s ).5.5.2.4.6.8.2.4.6.8 2 t [ms] Intuitively, more detail can be captured if the sampling rate is higher. If the highest frequency in a signal is f m, then it turns out that the process of sampling can be reversed without any loss, provided that F s > 2f m. The frequency 2f m is called the Nyquist rate or frequency. The sampling theorem states that a bandlimited waveform can be reconstructed exactly from its samples at rate F s, provided that F s is at least as large as the Nyquist rate. Digital computers can only work with discrete-time signals. Therefore, all signal processing operations in Matlab have to be performed on DT signals. It is often useful to choose a 6
sampling rate in Matlab that is much higher than the Nyquist rate and to pretend that the corresponding signal is a CT signal. This was done for the CT sinewave signal in the graph above. When the plot command in Matlab is used, then Matlab automatically connects adjacent samples by straight lines, thereby creating the illusion of a CT signal. But fundamentally, all signals in Matlab are DT signals that are stored in vectors and matrices with discrete indexes. Electrical Quantities 2 Decibels The decibel (db) is a logarithmic unit used for relative power measurements. A decibel is one tenth of a bel (B), a seldom used quantity named in honor of Alexander Graham Bell, the inventor of the telephone. Let P i and P o be two powers (input power P i and output power P o ) to be compared, e.g., the transmit and receive power in a wireless communication system. The power ratio in db with respect to P i is then expressed as G db = log ( P o P i ) db, where log stands for logarithm to base (log in Matlab). Thus, if a transmitter uses P i = W of transmit power and the receiver receives P o = nw, then G db = log( 9 /) = 9 db and we say that the signal is attenuated by 9 db. If an amplifier has a power gain of 2, i.e., P o = 2P i, then G db = log(2) = 3. 3 db and we say that the gain is 3 db. Through Ohm s law (V = R I or I = V/R and thus P = V I = V 2 /R) the input and output powers P i and P o are related to the input and output voltages V i and V o by P i = Vi 2 /R i and P o = Vo 2 /R o, where R i and R o are input and output resistances (often 5 Ω for wireless communication equipment). Thus, if R i = R o then G db can also be expressed in terms of the voltage ratio V o /V i as G db = 2 log ( V o V i ) db, where log again stands for logarithm to base. Note the factor of 2 instead of which is a consequence of the identity log(x 2 ) = 2 log(x). Thus, if an amplifier has a voltage gain of 2, then V o = 2 V i and G db = 2 log(2) = 6.2 6 db, i.e., a voltage gain of 2 corresponds to a power gain of 4 or, in decibels, to a gain of 6 db. Absolute powers are also often expressed in decibels with respect to an absolute reference power P i, e.g., with respect to P i = W (denoted by dbw) or with respect to P i = mw (denoted by dbm). An absolute transmit power of 2 W then corresponds to 3 dbw or 33 dbm. As another example, if a value of 7 dbm is specified for an electronic component to function properly, then this corresponds to an absolute power of 5. mw that the component needs. 7
3 Fourier Series Approximation in Matlab Let x(t) be a periodic CT waveform with period T and let x n = x(n T s ) be its DT representation with sampling rate F s = /T s. Assume that F s has been chosen large enough so that the sampling theorem is (approximately) satisfied (i.e., F s is greater or equal to two times the largest frequency in x(t)). Assume further that the total length of x n is N and that this corresponds (approximately) to the period T or an integer multiple of T. Then the Fourier series coefficients X k = T x(t) e j2πkt/t dt, T can be approximated by X k NT s N n= x n e j2πknts/(nts) T s = N N n= x n e j2πkn/n = X k N, k =,, 2,..., N, where X k are the discrete Fourier transform (DFT) coefficients of the DT sequence x n, i.e., X k = N n= x n e j2πkn/n, k =,, 2,..., N. If the blocklength N is a composite number, then there exist computationally efficient algorithms, collectively called fast Fourier transform (FFT) to compute X k. Thus, by choosing F s and N large enough the FS coefficients X k can be computed accurately and efficiently using the DFT coefficients X k that the FFT algorithms produce. In Matlab the X k can be computed and displayed using the following Matlab function: function showfs_v(xt,fs) %showfs_v Plot (approximation to) FS coefficients Xk of (periodic, % period N=length(xt)) waveform x(t) sampled at rate Fs, % Version. % Command format: showfs_v(xt,fs) N = length(xt); Xk = /N*fft(xt); ff = Fs/N*[:N-]; %Total number of samples (period) %Xk approximated using FFT %Frequency axis in Hz subplot(2) stem(ff,abs(xk),.-b ) grid ylabel( X_k ) str = FS Coefficient Approximation for x(t) ; str = [str, N= int2str(n), Fs= int2str(fs) Hz ]; str = [str, \Delta_f= num2str(fs/n) Hz ]; title(str) figure(gcf) 8
To test this function, a sinusoidal signal of the form x(t) = A cos(2πf t + θ ) + A 2 cos(2πf 2 t + θ 2 ), was generated. The result when A =, f = Hz, θ =, A 2 =.7, f 2 = 5 Hz, θ 2 = 9, F s = 8 Hz, and the total duration of x(t) is second, is shown in the figure below..5 FS Coefficient Approximation for x(t), N=8, Fs=8 Hz, f = Hz.4 X k.3.2. 2 3 4 5 6 7 8 4 Amplitude Modulation in Matlab 5 Filters A filter is an input-output system that exhibits frequency dependent behavior in the sense that specific frequency bands are passed through the system while others are rejected or at least attenuated significantly. An ideal lowpass filter (LPF) is a filter that passes all frequencies below a cutoff frequency f L without change and that rejects all frequencies above f L completely. An ideal highpass filter (HPF) performs the dual function of passing all frequencies above a cutoff frequency f H without change and completely rejecting all frequencies below f H. An ideal bandpass filter (BPF) with center frequency f c and bandwidth W completely rejects all frequencies below fc W/2 and above f c + W/2 and passes frequencies in the range f c W/2... f c + W/2 unchanged. The specification of real filters is more complicated because it is not possible in a finite amount of time to make an infinitely steep transition from the passband where the frequencies are passed without change to the stopband where the frequencies are rejected completely. The parameters that are typically used to specify real filters in terms of the magnitude of the frequency response are shown in the following figure. Note that ω = 2πf where ω is frequency in radians per second and f is frequency in Hertz. 9
LPF Magnitude Frequency Response Specification A p A p 3dB H(jω) in db R p R s ω<ω 3dB : Passband ω <ω<ω : Transition Band 3dB s ω <ω: Stopband s A s ω 3dB ω s ω > The passband extends from dc to the half-power or -3dB frequency ω 3dB. The magnitude response in the passband may have a maximum ripple of R p db associated with it. The stopband of the filter, which extends from ω s all the way to infinity, may also contain a ripple, but the attenuation must be at least R s db for all ω > ω s. Between ω 3dB and ω s lies the transition band in which frequencies are neither considered to be fully passing, nor to be fully rejected. In terms of the magnitude frequency specification, a filter which has a narrower transition band for a given filter order is considered to be better. 6 AM Transmitters 7 AM Receivers c 22, P. Mathys. Last revised: 2-5-2, PM. 2