Fourier Analysis of Signals and Systems Dr. Babul Islam Dept. of Applied Physics and Electronic Engineering University of Rajshahi

Outline Response of LTI system in time domain Properties of LTI systems Fourier analysis of signals Frequency response of LTI system

Linear Time-Invariant (LTI) Systems A system satisfying both the linearity and the time- invariance properties. LTI systems are mathematically easy to analyze and characterize, and consequently, easy to design. Highly useful signal processing algorithms have been developed utilizing this class of systems over the last several decades. They possess superposition theorem.

Linear System: System, T is linear if and only if + T System, T is linear if and only if i.e., T satisfies the superposition principle. + T

Time-Invariant System: A system T is time invariant if and only if T implies that T Example: (a) Since , the system is time-invariant. (b) , the system is time-variant.

Graphical representation of unit impulse. Any input signal x(n) can be represented as follows: 1 n 2 -1 -2 … Graphical representation of unit impulse. Consider an LTI system T. T Now, the response of T to the unit impulse is T Applying linearity properties, we have

Applying the time-invariant property, we have (LTI) LTI system can be completely characterized by it’s impulse response. Knowing the impulse response one can compute the output of the system for any arbitrary input. Output of an LTI system in time domain is convolution of impulse response and input signal, i.e.,

Properties of LTI systems (Properties of convolution) Convolution is commutative x[n]  h[n] = h[n]  x[n] Convolution is distributive x[n]  (h1[n] + h2[n]) = x[n]  h1[n] + x[n]  h2[n]

= Convolution is Associative: y[n] = h1[n]  [ h2[n]  x[n] ] = [ h1[n]  h2[n] ]  x[n] x[n] h2 h1 y[n] = x[n] h1h2 y[n]

Frequency Analysis of Signals Fourier Series Fourier Transform Decomposition of signals in terms of sinusoidal or complex exponential components. With such a decomposition a signal is said to be represented in the frequency domain. For the class of periodic signals, such a decomposition is called a Fourier series. For the class of finite energy signals (aperiodic), the decomposition is called the Fourier transform.

Fourier Series for Continuous-Time Periodic Signals: Consider a continuous-time sinusoidal signal, A Acos t This signal is completely characterized by three parameters: A = Amplitude of the sinusoid  = Angular frequency in radians/sec = 2f  = Phase in radians

Complex representation of sinusoidal signals: Fourier series of any periodic signal is given by: where Fourier series of any periodic signal can also be expressed as: where

Example:

Power spectrum of a CT periodic signal. Power Density Spectrum of Continuous-Time Periodic Signal: This is Parseval’s relation. represents the power in the n-th harmonic component of the signal. If is real valued, then , i.e., Power spectrum of a CT periodic signal. Hence, the power spectrum is a symmetric function of frequency.

Fourier Transform for Continuous-Time Aperiodic Signal: Assume x(t) has a finite duration. Define as a periodic extension of x(t): Therefore, the Fourier series for : where Since for and outside this interval, then

Now, defining the envelope of as Therefore, can be expressed as As Therefore, we get

Energy Density Spectrum of Continuous-Time Aperiodic Signal: This is Parseval’s relation which agrees the principle of conservation of energy in time and frequency domains. represents the distribution of energy in the signal as a function of frequency, i.e., the energy density spectrum.

Fourier Series for Discrete-Time Periodic Signals: Consider a discrete-time periodic signal with period N. Now, the Fourier series representation for this signal is given by where Since Thus the spectrum of is also periodic with period N. Consequently, any N consecutive samples of the signal or its spectrum provide a complete description of the signal in the time or frequency domains.

Power Density Spectrum of Discrete-Time Periodic Signal:

Fourier Transform for Discrete-Time Aperiodic Signals: The Fourier transform of a discrete-time aperiodic signal is given by Two basic differences between the Fourier transforms of a DT and CT aperiodic signals. First, for a CT signal, the spectrum has a frequency range of In contrast, the frequency range for a DT signal is unique over the range since

Second, since the signal is discrete in time, the Fourier transform involves a summation of terms instead of an integral as in the case of CT signals. Now can be expressed in terms of as follows:

Energy Density Spectrum of Discrete-Time Aperiodic Signal: represents the distribution of energy in the signal as a function of frequency, i.e., the energy density spectrum. If is real, then (even symmetry) Therefore, the frequency range of a real DT signal can be limited further to the range

Frequency Response of an LTI System For continuous-time LTI system For discrete-time LTI system

Conclusion The response of LTI systems in time domain has been examined. The properties of convolution has been studied. The response of LTI systems in frequency domain has been analyzed. Frequency analysis of signals has been introduced.