Lecture 3 ctft
- 1. 1 ECNG3025 Lecture 3 Fourier Analysis 1 The CTFT ECNG3025 © 2014 2 Use of Frequency Domain Analysis Telecommunication Channel |A| f fc Frequency Response Signal frequencies > fc are attenuated and distorted Communication channels Speech recognition The sound of ‘a’ spoken by 2 people |A| f SPEECH ANALYSIS RECOGNITION IDENTIFICATION
- 2. 2 ECNG3025 © 2014 3 Use of Frequency Domain Analysis The tone controls on stereo equipment Low freq - Bass High freq – Treble Harmonic Analysis in power systems Stock market fluctuation analysis Audiology Image processing And more….. ECNG3025 © 2014 4 Spectrum The frequency content of a signal is called Its spectrum Analogous to light Information about the frequency content of a signal is called Spectrum analysis For continuous signals can be done using a spectrum analyser
- 3. 3 ECNG3025 © 2014 5 Fourier Series Highlights A periodic signal can be expressed as an infinite sum of orthogonal functions. When these functions are the cosine and sine, the sum is called the Fourier Series. The frequency of each of the sinusoidal functions is an integer multiple of the fundamental frequency. Jean Baptiste Joseph Fourier ECNG3025 © 2014 6 Fourier Series Highlights Tp Original Signal xp(t) First 4 Terms of Fourier Series Sum of First 4 Terms of Fourier Series and xp(t) Original xp(t) First 4 Terms of Fourier Series Periodic signal expressed as infinite sum of sinusoids.
- 4. 4 ECNG3025 © 2014 7 Fourier Series Highlights The Fourier series has a number of common descriptions See ECNG 2011 Notes We used the complex exponential form: k tjk kectx 0 )( pT tjk p k dtetx T c 0 )( 1 where Recall that Euler’s relationships express e in terms of the sine and cosine functions ECNG3025 © 2014 8 Fourier Series Highlights Using the complex exponential form , we have: etc2,1,0,where )( 1 0 k dtetx T c T tjk k Complex Fourier coefficients
- 5. 5 ECNG3025 © 2014 9 Complex Fourier Coefficients kj kk ecc We can express the complex coefficients in this form The plot of |ck| versus angular frequency is called The Amplitude Spectrum of x(t) The plot of k versus angular frequency is called The Phase Spectrum of x(t) dtetx T c T tjk k 0 )( 1 The index, k, assumes integer values, so that the Amplitude and Phase spectra appear at discrete frequencies, k0. ECNG3025 © 2014 10 Example Recall the following example: The Fourier series coefficients for this function were shown to be: 2 a 2 a 2 T 2 T f(t) t T a k T a k ck sin
- 6. 6 -3 -2 -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 A sinc spectrum A plot of the magnitude of ck (normalised ie magnitude = 1) T a k T a k ck sin ECNG3025 11© 2014 Line spacing is a/T and the zero crossing points are at k= mT/a, m = - to (integer values) ECNG3025 © 2014 12 Observations -3 -2 -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 A sinc spectrum Periodic signal has a discrete spectrum Intuitive Think of a sinusoid What is its spectrum? What happens as period increases?
- 7. 7 ECNG3025 © 2014 13 Observations As T increases Line spaces decrease Spectral lines get closer together Zero crossing points move further apart Spectrum flattens Line spacing is a/T and the zero crossing points are at k= mT/a, m = - to (integer values) ECNG3025 © 2014 14 Fourier Series Limits FS applies to periodic signals How can it represent non-periodic waveforms such as speech? For example, a single spoken word is a non-periodic waveform because it does not repeat itself. What is the period of a non-periodic waveform?
- 8. 8 ECNG3025 © 2014 15 Non-periodic signal Solution is to treat it as a periodic waveform with an infinite period. If we assume that T tends towards infinity, then we can produce equations for non-periodic signals Remember our previous observations as T increased What is the period of an aperiodic signal? ECNG3025 © 2014 16 Non-periodic signal As the fundamental frequency approaches 0 a number of things happen 2 )( 0 0 0 0 f Cc k T k 0 0 )( 1 0 T tjk k dtetf T c T tj dtetfC )( 2 )( Thus Evolves to
- 9. 9 ECNG3025 © 2014 17 Non-periodic signal The harmonic designator k is dropped from the equations We now have an infinite number of k values ie k is now a continuous C() is therefore a continuous variable. But T is now infinite! Therefore: dtetfC tj )( 2 )( T tj dtetfC )( 2 )( ECNG3025 © 2014 18 Observations Periodic signal has a discrete spectrum From before Non-periodic signal has a continuous spectrum
- 10. 10 ECNG3025 © 2014 19 Fourier Transform With a bit of juggling, normalising and substitution: dtetxX tj )()( dtetfC tj )( 2 )( Called the Continuous Time Fourier Transform (CTFT) of x(t) ECNG3025 © 2014 20 Fourier Transform Of course there must be an Inverse Fourier Transform x t X e dj t ( ) ( ) 1 2
- 11. 11 ECNG3025 © 2014 21 Comments The Fourier Transform allows you to compute the frequency domain representation of a signal from the time domain signal The inverse Fourier Transform allows you to compute the time domain representation from the frequency domain signal ECNG3025 © 2014 22 Comments The CTFT and the inverse CTFT are known as the Fourier Transform pair They apply to non periodic signals The application to periodic signals will be covered in a later lecture.
- 12. 12 ECNG3025 © 2014 23 Continuous Time Fourier Transform (CTFT) Standard notation dtetxjXXtx tj )()(or)()}({F deXtxX tj )( 2 1 )()}({1 F •Note the similarity in form to the Laplace Transform. •Many of its properties are likewise similar. •There are several important exceptions.