1. TELE3113 Analogue and Digital Communications Review of Fourier Transform Wei Zhang w.zhang@unsw.edu.auSchool of Electrical Engineering and Telecommunications The University of New South Wales
2. Fourier TransformLet g(t) denote a nonperiodic deterministic signal, the Fouriertransform (FT) of the signal g(t) is given by ∞ G(f ) = g(t) exp(−j2πf t)dt. −∞The inverse Fourier transform is given by ∞ g(t) = G(f ) exp(j2πf t)df. −∞We call G(f ) and g(t) as the Fourier-transform pair, denoted by g(t) ⇔ G(f ). TELE3113 - Review of Fourier Transform. July 29, 2009. – p.1/1
3. SpectrumThe FT G(f ) is a complex function of frequency f , so it can beexpressed as G(f ) = |G(f )| exp(jθ(f )),where |G(f )| is called the amplitude spectrum of g(t); θ(f ) is called the phase spectrum of g(t). TELE3113 - Review of Fourier Transform. July 29, 2009. – p.2/1
4. Rectangular Pulse (1)Deﬁne rectangular function of unit amplitude and unit durationcentered at t = 0 as 1, −1 ≤ t ≤ 1 2 2 rect(t) = 0, t < − 1 or t > 1 2 2Then, a rectangular pulse of duration T and amplitude A, as tshown in Figure, can be expressed as g(t) = A rect( T ). g (t ) A t −T /2 0 T /2 TELE3113 - Review of Fourier Transform. July 29, 2009. – p.3/1
5. Rectangular Pulse (2)The FT of a rectangular pulse of duration T and amplitude A is t Arect ⇔ AT sinc(f T ) T sin(πλ)where sinc(·) denotes the sinc function as sinc(λ) = πλ . 1 0.8 0.6 0.4 sinc(λ) 0.2 0 −0.2 −0.4 −3 −2 −1 0 1 2 3 TELE3113 - Review of Fourier Transform. July 29, 2009. – p.4/1 λ
6. Properties of FT (1) Linearity Property: If g(t) ⇔ G(f ), then c1 g1 (t) + c2 g2 (t) ⇔ c1 G1 (f ) + c2 G2 (f ). Dilation Property: If g(t) ⇔ G(f ), then 1 f g(at) ⇔ G . |a| a TELE3113 - Review of Fourier Transform. July 29, 2009. – p.5/1
7. Properties of FT (2) Conjugation Rule: If g(t) ⇔ G(f ), then g ∗ (t) ⇔ G∗ (−f ). Duality Property: If g(t) ⇔ G(f ), then G(t) ⇔ g(−f ). TELE3113 - Review of Fourier Transform. July 29, 2009. – p.6/1
8. Properties of FT (3) Time Shifting Property: If g(t) ⇔ G(f ), then g(t − t0 ) ⇔ G(f ) exp(−j2πf t0 ). Frequency Shifting Property: If g(t) ⇔ G(f ), then exp(j2πfc t)g(t) ⇔ G(f − fc ). TELE3113 - Review of Fourier Transform. July 29, 2009. – p.7/1
9. Properties of FT (4) Modulation Theorem: Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then g1 (t)g2 (t) ⇔ G1 (f ) G2 (f ), ∞ where G1 (f ) G2 (f ) = −∞ G1 (λ)G2 (f − λ)dλ. Convolution Theorem: Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then g1 (t) g2 (t) ⇔ G1 (f )G2 (f ), ∞ where g1 (t) g2 (t) = −∞ g1 (τ )g2 (t − τ )dτ . TELE3113 - Review of Fourier Transform. July 29, 2009. – p.8/1
10. Properties of FT (5) Correlation Theorem: Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then ∞ g1 (τ )g2 (t − τ )dτ ⇔ G1 (f )G∗ (f ). ∗ 2 −∞ Rayleigh’s Energy Theorem: Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then ∞ ∞ |g(t)|2 dt = |G(f )|2 df. −∞ −∞ Note that in the above formula, it is “=”, not “⇔”. TELE3113 - Review of Fourier Transform. July 29, 2009. – p.9/1
11. LP versus BP Low-pass (LP) signal: Its signiﬁcant spectral content is centered around the origin f = 0. Band-pass (BP) signal: Its signiﬁcant spectral content is centered around ±fc , where fc is a constant frequency. TELE3113 - Review of Fourier Transform. July 29, 2009. – p.10/1
12. BandwidthDeﬁnition of bandwidth (BW): For LP signal, the BW is one half the total width of the main spectral lobe. For BP signal, the BW is the width of the main lobe for positive frequencies. TELE3113 - Review of Fourier Transform. July 29, 2009. – p.11/1
13. 3-dB Bandwidth 3-dB BW of the LP signal: the separation between zero frequency and the positive frequency at which the amplitude √ spectrum drops to 1/ 2 of the peak value at zero frequency. 3 dB Bandwidth of LP signal 1 0.9 −3 dB 0.8 0.7 BW 0.6 0.5 0.4 0.3 0.2 0.1 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 3-dB BW of the BP signal: the separation between the two √ frequencies at which the amplitude spectrum drops to 1/ 2 of the peak value at fc . TELE3113 - Review of Fourier Transform. July 29, 2009. – p.12/1
14. Dirac Delta FunctionThe Dirac delta can be loosely thought of as a function on thereal line which is zero everywhere except at the origin, where itis inﬁnite, +∞, x = 0 δ(x) = 0, x=0and which is also constrained to satisfy the identity ∞ δ(x) dx = 1. −∞ TELE3113 - Review of Fourier Transform. July 29, 2009. – p.13/1
15. Applications of δ Function dc Signal: 1 ⇔ δ(f ). Complex Exponential Function: exp(j2πfc t) ⇔ δ(f − fc ). Sinusoidal Function: 1 cos(2πfc t) ⇔ [δ(f − fc ) + δ(f + fc )]. 2 1 sin(2πfc t) ⇔ [δ(f − fc ) − δ(f + fc )]. 2j TELE3113 - Review of Fourier Transform. July 29, 2009. – p.14/1
16. ReferenceAll the proofs of the properties of FT are available inChapter 2 of the bookIntroduction to Analog & Digital Communications, 2nd Ed.by Simon Haykin and Michael Moher. TELE3113 - Review of Fourier Transform. July 29, 2009. – p.15/1
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