UNIT-II FOURIER TRANSFORM

Introduction We have seen that periodic signals can be represented with the Fourier series Can aperiodic signals be analyzed in terms of frequency components? Yes, and the Fourier transform provides the tool for this analysis

Introduction Contd. Time Discrete Fourier Periodic Series Transform Continuous Time Periodic Aperiodic

Fourier Transform in the General Case Given a signal x(t), its Fourier transform is defined as A signal x(t) is said to have a Fourier transform in the ordinary sense if the above integral converges

Existence of Fourier Transform The integral does converge if the signal x(t) is “well-behaved” and x(t) is absolutely integrable, namely, Note: well behaved means that the signal has a finite number of discontinuities, maxima, and minima within any finite time interval

Example: The DC or Constant Signal Consider the signal Clearly x(t) does not satisfy the first requirement since Therefore, the constant signal does not have a Fourier transform in the ordinary sense

Rectangular Form of the Fourier Transform Consider Since in general is a complex function, by using Euler’s formula

Fourier Transform of Real-Valued Signals If x(t) is real-valued, it is Moreover whence Hermitian symmetry

Inverse Fourier Transform Given a signal x(t) with Fourier transform , x(t) can be recomputed from by applying the inverse Fourier transform given by Transform pair

Properties of the Fourier Transform Linearity: Left or Right Shift in Time: Time Scaling:

Properties of the Fourier Transform Time Reversal: Multiplication by a Power of t: Multiplication by a Complex Exponential:

Properties of the Fourier Transform Multiplication by a Sinusoid (Modulation): Differentiation in the Time Domain:

Properties of the Fourier Transform Integration in the Time Domain: Convolution in the Time Domain: Multiplication in the Time Domain:

Properties of the Fourier Transform Parseval’s Theorem: Duality: if

Generalized Fourier Transform Fourier transform of Applying the duality property generalized Fourier transform of the constant signal

Generalized Fourier Transform of Sinusoidal Signals

Fourier Transform of Periodic Signals Let x(t) be a periodic signal with period T; as such, it can be represented with its Fourier transform Since , it is

Fourier Transform of the Unit-Step Function Since using the integration property, it is

Sampling Sampling is a continuous to discrete-time conversion. Most common sampling is periodic If a function x(t) contains no frequencies higher than B Hz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.

Types of Sampling

Sampling Theorem Statement: A continuous time signal can be represented in its samples and can be recovered back when sampling frequency fs is greater than or equal to the twice the highest frequency component of message signal. i. e. fs≥2fm.

Nyquist sampling rate for low-pass and band pass signals

Aliasing Effect Aliasing is a phenomenon where the high frequency components of the sampled signal interfere with each otherbecause of inadequate sampling fs  < 2fm . The overlapped region in case of under sampling represents aliasing effect, which can be removed by considering fs >2fm By using anti aliasing filters.

Fourier Transform Signal Analysis and Processing （1）Time Domain Analysis: t-A （2）Frequency Domain Analysis: f-A Fourier Transform 这里是否应该加入对频谱分析的一点解释呢？ in time-domain in frequency-domain In some situation, signal’s frequency spectrum can represent its characteristics more clearly.

？ Fourier Transform Signal Analysis and Processing: （1）Time Domain Analysis （2）Frequency Domain Analysis Fourier Transform is a bridge from time domain to frequency domain. Characteristic: continuous—discrete, periodic—nonperiodic . Type: Continuous periodic signals ？ Continuous nonperiodic signals Discrete nonperiodic signals Discrete periodic signals

Continuous non-periodic function’s Fourier Transform Time-domain Frequency-domain Conclusion : Continuous non-periodic function— Non-periodic continuous function

Discrete-time non-periodic sequence’s Fourier Transform Time-domain Frequency-domain Conclusion : Discrete non-periodic function— Continuous-time periodic function

Periodic  Discrete; NonperiodicContinuous Conclusion （1）Sampling in time domain brings periodicity in frequency domain. （2）Sampling in frequency domain brings periodicity in time domain. （3）Relationship between frequency domain and time domain Time domain Frequency domain Transform Continuous periodic Discrete non-periodic Fourier series Continuous non-periodic Continuous non-periodic Fourier Transform Discrete non-periodic Continuous periodic Sequence’s Fourier Transform Discrete periodic Discrete periodic Discrete Fourier Series Periodic  Discrete; NonperiodicContinuous

Basic idea of Discrete Fourier Transform In practical application, signal processed by computer has two main characteristics: (1) Discrete (2) Finite length Similarly, signal’s frequency must also have two main characteristics: (1) Discrete (2) Finite length But nonperiodic sequence’s Fourier Transform is a continuous function of , and it is a periodic function in  with a period 2. So it is not suitable to solve practical digital signal processing. Idea: Expand finite-length sequence to periodic sequence, compute its Discrete Fourier Series, so that we can get the discrete spectrum in frequency domain.

Discrete Fourier Transform-DFT Periodic sequence and its DFS

Discrete Fourier Transform-DFT Periodic sequence is infinite length. but only N sequence values contain information. Periodic sequence  finite length sequence. Relationship between these sequences? Infinite  Finite Periodic  Nonperiodic 周期序列和有限长序列存在本质上的联系

Discrete Fourier Transform-DFT Relationship between periodic sequence and finite-length sequence Periodic sequence can be seen as periodically copies of finite-length sequence. Finite-length sequence can be seen as extracting one period from periodic sequence. Finite-duration Sequence Periodic Sequence Main period

Properties of DTFT

Properties of DTFT

Convolution Property

Multiplication Property