Convolution and multiplication with mixtures of periodic and nonperiodic signals

Introduction

The FT and DTFT representations of periodic signals to analyze problems involving mixtures of periodic and nonperiodic signals. It is common to have the mixing of periodic and non-periodic signals in convolution and multiplication problems. For example, if a periodic signal is applied to a stable filter, the output is expressed as the convolution of the periodic input signal and the non-periodic impulse response. The tool we analyze problems involving mixtures of periodic and nonperiodic continuous-times signals is the pi. The DTFT applies to mixtures of periodic and non-periodic discrete-time signals. This analysis is possible as we now have FT and DTFT representations of bo periodic and non-periodic signals. 

Convolution of periodic and non-periodic signals

As we know, the convolution in the time domain corresponds to multiplication in the frequency domain. This property can be applied to problems in which one of the time-domain signals is periodic. Let that signal be x(t), so it will be represented as follows:

x(t) ←> X(jw) = 2pi Σ X[k] d(w-kw)

Here, X[k] are the Fourier series coefficients. We will substitute this representation into the convolution property to obtain:

y(t) = x(t) * h(t) ←→ Y(jw) = 2piΣ X[k]d(w-kw) H(jw) 

The form of Y( jw) corresponds to a periodic signal. Hence, y(t) is periodic with the same period as x(t) The most common application of this property is in determining the output of a filter with impulse response b(t) and periodic input x(t). 

Multiplication of periodic and non-periodic signals

Multiplication of g(t) with the periodic function x(t) gives a Fourier Transform consisting of a weighted sum of shifted versions of G(jw). As expected, the form of Y(jw) corresponds to the Fourier Transform of a non-periodic signal, since the product of periodic and non-periodic signals is non-periodic.

Fourier Transform representation of discrete-time signals

We will start by relating the FT to the DTFT. 

Let us consider the DTFT of an arbitrary discrete-time signal x[n]. We will have:

X(e^iΩ) = Σx[n] e^-iΩn

We need an Fourier Transform pair that will correspond to the DTFT pair x[n] ←→ X(e^iΩ). Substituting Ω = wT, we can obtain the following function of the continuous-frequency w:

Xs(jw) = X(e^iΩ)|Ω=wTs. 

Taking the inverse FT of X(jw), using linearity and the Fourier Transform pair:

d(t-nT) ←→ e^-jwTsn 

Will yield the continuous-time signal description:

x(t) = Σx[n]d(t-nT) 

Where x(t) is a continuous-time signal that corresponds tp x[n], while the Fourier transform X(jw) corresponds to the discrete-time Fourier transform X(e^iΩ). This representation has an associated sampling interval T that determines the relationship between continuous-time and discrete-time frequency. 

Sampling

Let x(t) be a continuous-time signal. We define a discrete-time signal x[n] that is equal to the “samples” of x(t) integer multiples of a sampling interval that is, x[n] = x[nT]. The impact of sampling is evaluated by relating the DTFT of x[n] to the FT. Our tool for exploring this relationship is the FT representation of discrete-time signals. 

It implies that we may mathematically represent the sampled signal as the product of the original continuous-time signal and an impulse train. The representation is commonly termed impulse sampling and is a mathematical tool used only to analyze sampling. 

The effect of sampling is determined by relating the FT of x(t) to the FT of x(t). Since multiplication in the time domain corresponds to convolution in the frequency domain, we have:

X(jw) = 1/2pi x(jw)* P(jw). 

Substituting the value for P(jw) determined before into this relationship, we can obtain: 

X(jw) = 1/2pi X(jw) * 2pi/ T Σ d(w-kw). 

Where, w = 2pi/T is the sampling frequency. 

Therefore, the FT of the sampled signal is given by an infinite sum of shifted versions of the FT. The shifted versions are offset by integer multiples of w. The shifted versions of X(jw) may overlap with each other if w is not large enough compared to the frequency extent, or the bandwidth, of X(jw). The frequency band is said to lie in the range -W < w < W. The T increases and w decreases, alongside the replicas of X(jw) move closer together, finally overlapping one another.

Overlap in the shifted replicas of the original spectrum is termed aliasing. Aliasing refers to the phenomenon of a high-frequency discrete-time component. It distorts the spectrum of the sampled signal. The replicas add, and therefore, the basic shape of the spectrum changes from portions of a triangle to a constant. 

Reference

Convolution and multiplication with mixtures of periodic and nonperiodic signals