Detecting a frequency swept sinusoid and its parameters?
You could compute the instantaneous frequency of the signal in the frame. This can be done as outlined in this paper (e.g. Eq.(9)). You need to compute the analytic signal using a Hilbert transformer:

$$x_a[n]=x[n]+jx_h[n]\tag{1}$$

where $x_h[n]$ is the Hilbert transform of $x[n]$. The instantaneous frequency is given by the (discrete-time) derivative of the phase of (1). If you use a first order difference for approximating the discrete-time derivative you'll get the formula given in Eq. (9) of the paper cited above. Given an estimate of the instantaneous frequency, you could simply fit a line to estimate the sweep parameter.

This little Matlab/Octave script shows how this approach would work with a toy problem:

% create chirp signal
N = 256;
n=(0:N-1)';
fx=(0.1+0.01*n/N);
x=sin(2*pi*fx.*n);

% compute analytic signal
xh=hilbert(x);
xr=real(xh);xi=imag(xh);

% compute instanteneous frequency (could be done over fewer points)
tmp1 = xr(1:N-1).*xi(2:N) - xr(2:N).*xi(1:N-1);
tmp2 = xr(1:N-1).*xr(2:N) + xi(1:N-1).*xi(2:N);
f=atan2(tmp1,tmp2)/(2*pi);  % (biased) estimate of instanteneous frequency

% linear regression over center values of f
skip = floor(N/4);    % skip unreliable points at beginning and end of f
na = skip:(N-skip);
A=[ones(length(na),1),na(:)];
u=A\f(na);

f0=u(1); k=u(2)/2;  % remove bias
[f0,k]

% reconstructed linear frequency
fr = f0 + k*n;

plot(n,fx,(2:N),f-k*(2:N)',n,fr,':')
legend('exact','estimated','reconstructed')
axis([0,N,f0*.9,(f0+k*N)*1.1])


The plot shows the actual linear frequency of the signal (blue), the estimate from the analytic signal (green), and the reconstructed frequency obtained from fitting a line through the estimated frequency. Note that the estimate is unreliable at both ends, so the line is fitted to the center values of the frequency estimate.