Overview of Fourier Representation Properties Review of Signal Types • Range of equations (finite versus infinite) DT Periodic DT Fourier Series CT Periodic CT Fourier Series • Signal and transform classes (periodic versus nonperiodic) DT Nonperiodic DT Fourier Transform • Linearity CT Nonperiodic CT Fourier Transform • Time and frequency shifts • We have discussed two categories of signals • Symmetry of transforms – Continuous-time & discrete-time • Transform relationship to signal symmetry – Periodic and nonperiodic • Amplitude/phase representations • Each combination of these properties has an appropriate transform • Time and frequency scaling • Keep in mind – Periodic signals are power signals • Convolution: nonperiodic versus periodic signals – The CTFT and DTFT usually only converge for energy signals • Multiplication – Can be applied to periodic and almost periodic signals if we • Parseval’s theorem allow the CTFT and DTFT to include impulses • Duality J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 1 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 2 Fourier Series & Transform Summary Fourier Series & Transform Observations • Periodic signals can be represented as a sum of sinusoids X [ k ] = 1 � X [ k ] e jk Ω o n � x [ n ]e − jk Ω o n DTFS x [ n ] = • Nonperiodic signals can be represented as an integral of sinusoids N k = <N> n = <N> • DT signals ∞ X [ k ] = 1 � x ( t )e − jkω o t d t � X [ k ] e jkω o t – Can be represented by a finite frequency range CTFS x ( t ) = T T – Have transforms that are periodic k = −∞ + ∞ x [ n ] = 1 � X (e j Ω ) = X (e j (Ω+2 π ) ) X (e jω ) e j Ω n dΩ X [ k ] = X [ k + N ] � X (e jω ) = x [ n ] e − j Ω n DTFT 2 π 2 π n = −∞ – This is a consequence of e j Ω n = e j (Ω ± ℓ 2 π ) n � + ∞ � + ∞ x ( t ) = 1 X ( jω ) e jωt d ω x ( t ) e − jωt d t CTFT X ( jω ) = • CT signals 2 π −∞ −∞ – Must be represented by an infinite frequency range , in general – Have transforms that are nonperiodic, in general • Each transform has a synthesis and analysis equation X [ k ] � = X [ k + N ] X ( jω ) � = X ( j ( ω + ω o )) • Each transforms the signal (function of time) into a function of – This is because e jω 1 t � = e jω 2 t unless ω 1 = ω 2 frequency J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 3 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 4
Notes on Terminology Property 1: Linearity � FS • Many of the transforms share the same properties ⇐ ⇒ a 1 X 1 [ k ] + a 2 X 2 [ k ] a 1 x 1 [ n ] + a 2 x 2 [ n ] • The following slides cover the primary properties of the transforms FT a 1 X 1 (e jω ) + a 2 X 2 (e jω ) ⇐ ⇒ • I will use the word transform generally to refer to both the Fourier � FS ⇐ ⇒ a 1 X 1 [ k ] + a 2 X 2 [ k ] series transforms and the Fourier transforms a 1 x 1 ( t ) + a 2 x 2 ( t ) FT ⇐ ⇒ a 1 X 1 ( jω ) + a 2 X 2 ( jω ) • The book avoids this and uses the word representation • All of the Fourier transforms are linear • This follows directly from the linearity property of sums and integrals • Note that for periodic signals, both components must have the same fundamental period J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 5 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 6 Example 1: Linearity Application Example 1: Applying the Definition Derive one of the four linearity relationships given on the previous slide. Suppose two people sing into a microphone at the same time. Under what conditions is it possible to design a linear time-invariant filter that separates one of the voices? What characteristics would you want the filter to have? J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 7 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 8
Application Example 1: Workspace Application Example 2: Applying the Definition Suppose you obtain an old recording of a song that you like that is corrupted with a constant humming background noise. Is it possible to design an LTI filter that will eliminate the humm? What characteristics would you want the filter to have? How much would it distort the original signal? J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 9 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 10 Application Example 2: Workspace Property 2: Time Shift � FS e − jk Ω o n o X [ k ] ⇐ ⇒ x [ n − n o ] FT e − j Ω n o X (e j Ω ) ⇐ ⇒ � FS e − jkω o t o X [ k ] ⇐ ⇒ x ( t − t o ) FT e − jωt o X ( jω ) ⇐ ⇒ • What effect does a time shift have on even/odd symmetry in general? • What effect does a time shift have on phase? • Suppose the phase of the Fourier representation is zero at all frequencies. What is the effect of shifting the signal? • What effect does a time shift have on amplitude? J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 11 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 12
Example 2: Time Shift Example 2: Workspace Derive one of the four time-shift relationships on the previous slide. J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 13 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 14 Application Example 3: Speaker Recognition Application Example 3: Workspace Suppose you are asked to design a security system that only allows a single authorized person to open a door. To confirm the person’s identity, they must press a button and speak a password. • How would you design such a system? • How would you account for the variable delay between the start of the signal (when they push the button) and when they speak the word? • How would you prevent an intruder from entering even if they say the same password? J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 15 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 16
Application Example 3: Workspace Property 3: Frequency Shift FS e jk s Ω o n x [ n ] ⇐ ⇒ X [ k − k s ] FT e j Ω s n x [ n ] ⇒ X (e j (Ω − Ω s ) ) ⇐ FS e jk s ω o t x ( t ) ⇐ ⇒ X [ k − k s ] FT e jω s t x ( t ) ⇐ ⇒ X ( j ( ω − ω s )) • Ω s and ω s is the frequency of the modulating complex sinusoid • For periodic signals the modulating complex sinusoid must be an integer multiple of the fundamental frequency, k s Ω o (why?) • What effect does multiplying (modulating) a signal with a complex sinusoid have? J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 17 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 18 Example 3: Frequency Shift Example 3: Workspace Derive one of the four frequency-shift relationships on the previous slide. J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 19 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 20
Application Example 4: Communications Channel Example 4: Workspace Suppose you wish to transmit a stereo signal over a single telephone line. You can model a telephone line as a lowpass filter with a 4 kHz passband. • How can you combine the two signals into a single signal in such a way that you can extract the original signals? • What must you compromise in combining the two signals into a single signal? • What properties must the signals have in order to send them over the telephone line without loss? J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 21 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 22 Example 4: Workspace Property 4: Transform Symmetry X [ − k ] = X ∗ [ k ] X (e − j Ω ) = X ∗ (e jω ) X ( − jω ) = X ∗ ( jω ) • Each transform can be used to synthesize the signal as a sum or integral of complex sinusoids • The resulting signal is real • Thus all the imaginary components must cancel (sum or integrate to zero) • This results in the complex-conjugate symmetry of the coefficients J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 23 J. McNames Portland State University ECE 223 Fourier Properties Ver. 1.11 24
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