Signal and System Analysis Objectives: To define the energy and power of a signal To classify different types of signals To introduce special functions commonly used in telecom. To define linear and time-invariant systems To define convolution To introduce Fourier series and Fourier transform To explain the concept of negative frequency To show how the signal may be described in either the time domain or the frequency domain and establish their relationship. To study autocorrelation and power spectral density

Classification of Signals As mentioned before, a signal represents the message that is to be sent across the channel. Let’s start looking into more detail. Signals can be classified in various ways: Continuous- or discrete-time signals Signals associated with a computer are discrete-time signals

Classification of Signals Periodic and nonperiodic signals A periodic signal is one that repeats itself exactly after a fixed length of time. g(t) = g(t + T)for all t g[n] = g[n+N] for all n The smallest positive number T (or N) that satisfies the above equation is called the period.

Classification of Signals Deterministic and random signals Deterministic signal: can be mathematically characterized completely in the time domain. Random signal: specified only in terms of probabilistic description All signals encountered in telecommunications are random signals. If a message is used to convey information, it must have some uncertainty (randomness) about it. Otherwise, why communicate?

We know,power Energy For our purpose, we will neglect the Energy and average power of a signal and Power and energy of a signal

Some important signals Singularity functions in continuous-time systems Singularity functions are discontinuous or have discontinuous derivatives. Singularity functions are mathematical idealizations and, strictly speaking, do not occur in physical systems. They serve as good approximations to certain limiting conditions in physical systems. We will discuss two types of singularity functions: unit step function and unit impulse function

Step function Unit step function The unit step function is defined as (u(t) has no definition at t = 0, or one may define u(0) = 1 or u(0) = ½)

Impulse function Unit impulse function The unit impulse function is defined as

Relationship between Relationship between  (t) and u(t)

Multiplication of a function by  (t) f(t)  (t) = f(0)  (t),f(t) continuous at t=0. f(t)  (t – T) = f(T)  (t – T),f(t) continuous at t=T. Sampling property of  (t),  [n] Further properties of Further properties of  (t),  [n]

Exponential functions Euler’s formula Exponential functions are important class of functions in this course. We will be making use of this fact a lot in this course

Systems A system is defined as a set of rules that maps an input signal to an output signal. g(t) -- input signal (or source signal); y(t) -- output signal (or response signal); Input and response are represented as g(t)  y(t) and read as input g(t) causes a response y(t).

Some properties of systems Linear and nonlinear systems For a system with g 1 (t)  y 1 (t) and g 2 (t)  y 2 (t), A system is said to be linear if the following properties hold: 1)ag 1 (t)  ay 1 (t) 2)ag 1 (t) + bg 2 (t)  ay 1 (t) +by 2 (t) for any scalar a,b Otherwise, the system is nonlinear. Examples

Time-invariant and time-varying systems A system is time-invariant if a time shift in the input results in a corresponding time shift in the output g(t – t 0 )  y(t – t 0 )for any t 0. i.e. when the same input is applied to a system today or tomorrow, the output is the same, just shifted in time accordingly Some properties of systems

Exercise Any system not meeting this requirement is said to be time- varying. Example: Another example: Humans We will focus on Linear and Time-Invariant (LTI) systems in this course Time Invariance

Exercise Non-linear, time-invariant Linear, time-variant Non-linear, time-invariant Linear, time-invariant

Convolution in LTI systems Consider a discrete-time LTI system. If we apply the impulse function as the input, let be the output. We call the Impulse response of an LTI system Now, since the system is linear, if we apply a scaled version of the impulse,

Convolution in LTI systems Furthermore, since the system is time-invariant, if we apply a delayed version of the impulse, Now, what if we have this?

Convolution in LTI systems If we define Do you see a pattern? In an LTI system with input and impulse response, Similarly for continuous-time systems,

Convolution in LTI systems We write convolution as Example:

Convolution in LTI systems In an LTI system, the impulse response or completely characterize the system. LTI system

Fourier Analysis Fourier analysis Alternative Representation of a periodic signal A signal can be represented in either the time domain (where it is a waveform as a function of time) or in the frequency domain. Such representation is called the spectrum of the original time-domain signal. If the signal is specified in the time domain, we can determine its spectrum, and vice versa. Fourier analysis provides a link between the time domain and the frequency domain. Amplitude

Fourier Analysis Claim: A periodic function g(t) of period T, can be expressed as an infinite sum of sinusoidal waveforms with frequency This summation is called Fourier series. Fourier series can be written in several forms. One such form is the trigonometric Fourier series: The constants are called Fourier Coefficients

Obtaining the Fourier Coefficients

Fundamental Frequency and Harmonics Given with period T, in the formulation is referred to as the fundamental frequency and the others are called harmonics of g(t)

Example

Fourier serious approximation to periodic functions In practice, As, the Fourier series approaches the original function

Sinc function Rewrite: If we define a function and define sinc(0)=1, we may rewrite This is called the sinc function

Sinc function

Exponential Fourier series This is called the exponential Fourier series Since and we can rewrite as where  0 = 2  /T

Exponential Fourier series How are the coefficients in the exponential Fourier series c n related to a n, b n ? where c 0 = a 0 / 2, c n = (a n – jb n ) / 2, c -n = (a n + jb n ) / 2 Exponential Fourier series

Fourier coefficients in complex fourier series Fourier coefficients c n in complex fourier series c n can be complex. | c n | is called spectral amplitude and represents the amplitude of n th harmonic. Arg(c n ) is known as the spectral phase Graphic representation of spectral amplitude along with the spectral phase is called complex frequency spectrum of the original signal g(t) Since c 0 = a 0 / 2, c n = (a n – jb n ) / 2, c -n = (a n + jb n ) / 2,

Negative Frequency ~!?!? Since n takes on negative values, apprently there is “negative frequency”. But remember, this exists only because we want to express and as In reality, frequencies can only have positive values. However, it is mathematically easier to use exponential representation rather than trigonometric. That’s why we allow the existence of negative frequency. Keep this in mind ~

A bit of revision Up to now, we have shown that a periodic function in time g(t) can be specified in two equivalent ways: Time domain representation -- waveform. Frequency domain representation – spectrum, Fourier coefficients. If the signal is specified in time domain, we can determine its spectrum and vice versa.

Spectrum dependence on period of signal

When T is larger, becomes smaller, the spectrum becomes denser. When T goes to infinity, Only a single pulse of width  in the time domain.  0  0, i.e., no spacing is left between two line-components; Thus, the spectrum becomes continuous and exists at all frequencies. (However, there is no change in the shape of the envelope of the spectrum). Spectrum dependence on period of signal

Fourier Transform The Fourier transform of a signal g(t) is defined by and g(t) is called the inverse Fourier transform of G(  ) The functions g(t) and G(  ) constitute a Fourier transform pair: g(t)  G(  ) G(  ) = F[g(t)]andg(t) = F -1 [G(  )] What is the difference between Fourier transform and Fourier series?

Fourier Transform Fourier transform is different from the Fourier Series in that its frequency spectrum is continuous rather than discrete. Fourier transform is obtained from Fourier series by letting T   (for a nonperiodic signal). The original time function can be uniquely recovered from its Fourier transform.

Fourier Transform and Fourier Series Please keep in mind that A periodic signal spectrum has finite amplitudes and exists at discrete set of frequencies. Those amplitudes are also called the Fourier coefficients of the periodic signal A non-periodic signal has a continuous spectrum G(  ) and exist at all frequencies.

Fourier transform of some useful functions Rectangular function: Proof

Unit impulse function:  (t)  1and 1  2  (  ) Proof Fourier transform of some useful functions

Sinusoidal function cos(  0 t) cos(  0 t)   [  (  +  0 ) +  (  -  0 )] Proof Fourier transform of some useful functions

Properties of Fourier Transform Linearity property If g 1 (t)  G 1 (  )andg 2 (t)  G 2 (  ) thena 1 g 1 (t) + a 2 g 2 (t)  a 1 G 1 (  ) + a 2 G 2 (  ) where a 1 and a 2 are constants This property is proved easily by linearity property of integrals used in defining Fourier transform

Symmetry property If g(t)  G(  ),thenG(t)  2  g(-  ) Proof we can interchange the variable t and , i.e. let t  ,   t, then Properties of Fourier Transform

Time scaling property let x = at, then dt = dx/a, case 1:when a > 0, case 2:when a < 0, then t   leads to x  - , Combined, the two cases are expressed as, Proof Properties of Fourier Transform

Important Observation: Time domain compression of a signal results in spectral expansion Time domain expansion of a signal results in spectral compression Properties of Fourier Transform

Time shifting property put t – t 0 = x, so that dt = dx, then Proof Frequency shifting property Properties of Fourier Transform

Significance Multiplication of a function g(t) by exp(j  0 t) is equivalent to shifting its Fourier transform in the positive direction by an amount  Frequency translation theorem. Translation of a spectrum helps in achieving modulation, which is performed by multiplying the known signal g(t) by a sinusoidal signal. Therefore, Properties of Fourier Transform

The multiplication of a time function with a sinusoidal function translates the whole spectrum G(  ) to  0. exp(j  0 t) can also provide frequency translation, but it is not a real signal. Hence, sinusoidal function is used in practical modulation system. Modulation Theorem

Convolution Suppose that g 1 (t)  G 1 (  ) and g 2 (t)  G 2 (  ), then, what is the waveform of g(t) whose Fourier transform is the product of G 1 (  ) and G 2 (  )? This question arises frequently in spectral analysis, and is answered by the convolution theorem. The convolution of two time function g 1 (t) and g 2 (t), is defined by the following integral Properties of Fourier Transform

Convolution Theorem Time convolution theorem If g 1 (t)  G 1 (  ) and g 2 (t)  G 2 (  ) Then g 1 (t) * g 2 (t)  G 1 (  )G 2 (  ) Frequency convolution theorem If g 1 (t)  G 1 (  ) and g 2 (t)  G 2 (  ) Then The proof is similar to time convolution theorem.

Convolution Theorem: Applications g 1 (t) * g 2 (t)  G 1 (  )G 2 (  ) If we let, then But (time shifting property) Therefore, convolving with a delta function shifted in time by corresponds to a shift of the original signal by

Signal transmission through a linear system y(t) = g(t) * h(t) when g(t)  G(  ), h(t)  H(  ), y(t)  Y(  ), h(t) is the impulse response, i.e. if the input is  (t), then y(t) = h(t). By convolution theorem Y(  ) = G(  )H(  ) where H(  ) is the system transfer function.

Signal Analysis Signal power Signal-to-noise ratio (S/N) is an important parameter used to evaluate the system performance. Noise, being random in nature, cannot be expressed as a time function, like deterministic waveform. It is represented by power. Hence, to evaluate the S/N, it is necessary to evolve a method for calculating the signal power. For a general time domain signal g(t), its average power is given by

Signal Analysis For a periodic signal, each period contains a replica of the function, and the limiting operation can be omitted as long as T is taken as the period. For a real signal Example Find the power of a sinusoidal signal cos  0 t. Solution Is it also possible to determine the signal power in frequency domain?

Frequency domain representation for signals of arbitrary waveshape When dealing with deterministic signals, knowledge of the spectrum implies knowledge of the time domain signal. For an arbitrary (random) signal, Fourier analysis cannot be used because g(t) is not known analytically. For such an undeterministic signal (which include information signals and noise waveforms), the power spectrum S g (  ) (or power spectral density) concept is used. Signal Analysis

The power spectrum describes the distribution of power versus frequency. The average signal power is then given by where S g (  )  0 for all . Another way to evaluate the signal power! Signal Analysis

Correlation Correlation measure of similarity between one waveform, and time delayed version of the other waveform. The autocorrelation function is a special case of convolution, and it measures the similarity of a function with its delayed replica, and is given by

Signal Analysis Important properties of autocorrelation (1) the autocorrelation for  = 0 is average power of the signal The third way to evaluate signal power! (2) power spectral density S g (  ) and autocorrelation function of a power signal are Fourier transform pair

Questions (Signal Analysis) 1.What type signal is the most fundamental? 2.How do you define a periodic signal? 3.Does a periodic signal exist within a limited time period? 4.Is the message signal a deterministic signal? 5.Does negative frequency physically exist? 6.What equipments you are going to use in order to observe the signal waveform and spectrum? 7.How does a pulsed signal differ from a sinusoidal signal? 8.What is the frequency domain description of a signal? Is it more or less useful than the time domain?

Questions (Signal Analysis) 9. Suppose that g 1 (t)  G 1 (  ) and g 2 (t)  G 2 (  ), what is the waveform of g(t) whose Fourier transform is the product of G 1 (  ) and G 2 (  )? 10. How do you measure the similarity between the signal and its delayed replica? 11. How many methods you know for signal power calculation? *12. What is the significance of the time- and frequency- scaling property of Fourier Transform?

Exercise Problems (Signal Analysis) 1. Evaluate the integrals 2. Simplify the following expressions: (a) [sint/(t + 2)]  (t);(b) [1/(j  +2)]  (  + 3); (c) [sin(k  )/  ]  (  ); 3. Calculate the (a) average value, (b) ac power, and (c) average power of the periodic waveform v(t) = 1 + cos  0 t.

Exercise Problems (Signal Analysis) 4. Prove that 5. If g(t)  G(  ), then show that g*(t)  G*(-  ). 6. Find the Fourier transform of the signal f(t) = [A + f m (t)]cos  c t if f m (t) has a spectrum F m (  ). 7. If f(t) has a spectrum F(  ), find the Fourier transform of the following functions: (a) f(t/2 – 5);(b) f(3 – 3t);(c) f(2 + 5t); 8. Determine the average power of the following signals: (a) Acos  0 t + B sin  0 t;(b) (A + sin  0 t) cos  0 t;

Exercise Problems (Signal Analysis) *9. Find the autocorrelation function of the signal, g(t) = Ecos  0 t; 10. For a power signal, g(t) = Acos(200  t)cos(2000  t), determine the average power.

Math. Table Trigonometric Identities Selected Fourier Transform Pairs

Math. Table Properties of Fourier Transform Linearity: a 1 g 1 (t) + a 2 g 2 (t)  a 1 G 1 (  ) + a 2 G 2 (  ) Symmetry: If g(t)  G(  ),thenG(t)  2  g(-  ) Time scaling: Time shifting: Frequency shifting: Modulation theorem: Time convolution: g 1 (t) * g 2 (t)  G 1 (  )G 2 (  ) Frequency convolution: Conjugate functions:g*(t)  G*(-  ) Time differentiation: Time integration:

Signal Analysis Classification of signals: Signals can be classified in various ways which are not mutually exclusive: Continuous (analog) and discrete (digital) signals: Continuous signals are those that do not have any discontinuity in the time domain. Discrete signals are those that assume only specific values at a certain time (and thus have discontinuities). Information-carrying signals can be either continuous or discrete. e.g. signals associated with a computer are digital because they take on only two values (binary signals)

Signal Analysis Therefore, all signals we have to deal with in telecommunications are random nonperiodic signals in reality. However, frequently we will use deterministic periodic signals to demonstrate a point because they are much easier to work with mathematically.

Signal Analysis Communication systems may involve complex waveforms, it is desirable to revolve them in terms of sinusoidal functions. Signal analysis is a tool for achieving this aim. Principle of signal analysis: To break up all the signals into summations of sinusoidal components.  A given signal can be described in terms of sinusoidal frequencies.

Signal Analysis The amplitude plot of F n is a discrete spectrum existing at  = 0,  0,  2  0,  3  0, …, have amplitudes A  /T, (A  /T)Sa(  /T), (A  /T)Sa(2  /T), …, etc. respectively.

Signal Analysis Interesting phenomena

If the input is a delta function at t = , i.e. it is  (t   ), then the output is h(t   ) and This means that, convolving a pulse x(t) located near t = 0 with a delta function located at t =  has the effect of shifting x(t) to around t = . This also applies in the frequency domain, and is shown schematically below. _