Chapter 2. Fourier Representation of Signals and Systems

2.3 The Inverse Relationship Between Time and Frequency If the time-domain description of a signal is changed, the frequency-domain description of the signal is changed in an inverse manner, and vice versa. If a signal is strictly limited in frequency, the time-domain description of the signal will trail on indefinitely, even though its amplitude may assume a progressively smaller value. A signal cannot be strictly limited in both time and frequency.

2.3 The Inverse Relationship Between Time and Frequency Bandwidth A measure of extent of the significant spectral content of the signal for positive frequencies. Commonly used three definitions Null-to-null bandwidth When the spectrum of a signal is symmetric with a main lobe bounded by well-defined nulls ? we may use the main lobe as the basis for defining the bandwidth of the signal 3-dB bandwidth Low-pass type : The separation between zero frequency and the positive frequency at which the amplitude spectrum drops to 1/√2 of its peak value. Band-pass type : the separation between the two frequencies at which the amplitude spectrum of the signal drops to 1/√2 of the peak value at fc. Root mean-square (rms) bandwidth The square root of the second moment of a properly normalized form of the squared amplitude spectrum of the signal about a suitably chosen point. It lends itself more readily to mathematical evaluation than the other two definitions of bandwidth Although it is not as easily measured in the lab.

2.3 The Inverse Relationship Between Time and Frequency Time-Bandwidth Product The produce of the signal’s duration and its bandwidth is always a constant Whatever definition we use for the bandwidth of a signal, the time-bandwidth product remains constant over certain classes of pulse signals Example

2.4 Dirac Delta Function The theory of the Fourier transform is applicable only to time functions that satisfy the Dirichlet conditions To combine the theory of Fourier series and Fourier transform into a unified framework, so that the Fourier series may be treated as a special case of the Fourier transform To expand applicability of the Fourier transform to include power signals-that is, signals for which the condition holds. Ex) g(t)=1.0

2.4 Dirac Delta Function Dirac delta function Having zero amplitude everywhere except at t=0, where it is infinitely large in such a way that it contains unit area under its curve. Sifting property : Replication property :

2.4 Dirac Delta Function

2.4 Dirac Delta Function Example 2.10 The Delta Function as a Limiting Form of the Gaussian Pulse

2.4 Dirac Delta Function Applications of the Delta Function 1. Dc signal By applying the duality property to the Fourier transform pair of Eq.(2.65) A dc signal is transformed in the frequency domain into a delta function occurring at zero frequency 2. Complex Exponential Function By applying the frequency-shifting property to Eq. (2.67)

2.4 Dirac Delta Function 3. Sinusoidal Functions

2.4 Dirac Delta Function 4. Signum Function This signum function does not satisfy the Dirichelt conditions and therefore, strictly speaking, it does not have a Fourier transform The double exponential pulse The limiting form

2.4 Dirac Delta Function

2.4 Dirac Delta Function 5. Unit Step Function

2.4 Dirac Delta Function 6. Integration in the time Domain (Revisited) Without the assumption of G(0)=0

2.5 Fourier Transform of Periodic Signals A periodic signal can be represented as a sum of complex exponentials Fourier transforms can be defined for complex exponentials Consider a periodic signal gT0(t) Complex Fourier coefficient f0 : fundamental frequency

2.5 Fourier Transform of Periodic Signals Let g(t) be a pulselike function

2.5 Fourier Transform of Periodic Signals One form of Possisson’s sum formula and Fourier-transform pair Fourier transform of a periodic signal consists of delta functions occurring at integer multiples of the fundamental frequency f0 and that each delta function is weighted by a factor equal to the corresponding value of G(nf0). Periodicity in the time domain has the effect of changing the spectrum of a pulse-like signal into a discrete form defined at integer multiples of the fundamental frequency, and vice versa.

2.5 Fourier Transform of Periodic Signals Example 2.11 Ideal Sampling Function