Brown and Yule (1983) take the analysis of discourse to be "the analysis of language in use. As such, it cannot be restricted to the description of linguistic forms independent of the purposes or functions which those forms are designed to serve in human affairs" (p. 1). If we admit that discourse analysis is the study of the use of language that serves "human affairs", therefore culture is the broadest context in which human affairs take place.

The method of discourse analysis was originally developed by Michel Foucault. Discourse analysis in Foucault's sense is actually not a "method" but rather a technique or rather an instrument that Foucault tried to correct via his objects of investigation. He applied discourse analysis when he found that the objects he was studying, as defined by him, were not quite as he had imagined. He describes the path of discourse analysis as groping forward from "book to book." Discourse analysis is not to be confused with hermeneutics. It is even more likely to be a qualitative-empirical approach to research (but more like a chimera something in between). But discourse analysis does have parts of hermeneutics. But hermeneutics tries to understand - discourse analysis does not.

One could understand the discourse analysis rather scientifically, it analyzes like a natural science. According to Foucault, sense of objects, sense of something, sense of things is bound to conditions and prerequisites.

Foucault's interest was also in history. He wanted to understand regularities and events at a specific time, to know how they became possible. For him, reality (also culture) is something historically determined. Foucault wants to hear what is silent. Voices "that are to be silenced" were of particular interest to him. Discourse analysis is a way to make these voices of the vulnerable subject audible.

In the meantime, however, there are a number of further developments in Foucault's discourse analysis, especially under the empirical paradigm. The dispositive analysis could also be interesting if you are more likely to examine institutions in cultures.

Discourse analysis always focuses on culture. There is a connection between discourse analysis and culture. Foucault not only wanted to dig up historical sources, but also to understand changes and developments in culture. He was interested in the things that did not prevail culturally. He wanted to reconstruct cultural power technologies of the respective time.

I agree with both Zouheir Maalej and Pierre-Carl Link. Discourse analysis is primarily intertwined with language and culture. Michel Foucault's viewpoints about discourse are prominent and numerous scholars within sociolinguistics, sociology, and cultural studies and in the broader field of social sciences have contributed to conceptualizing discourse analysis (e.g., Norman Fairclough, Anthony Giddens, among others).

With respect to the relationship between discourse and culture, it is depending on the field and context, but one of the crucial points of discourse analysis is to increase an in-depth understanding of public dialogues, debates, arguments, and communications, or popular inferences and assumptions about a specific topic or event function. Thus, scholars consider various types of methods such as historical narratives, media representations, and oral testimonies as empirical evidence to communicate with audiences. Overall, discourse analysis entails written and textual data and at times demands hermeneutics to interpret specific themes as discourse.

In addition to the points laid out by others thus far, it’s worth emphasising that contemporary scholars (e.g. Blommaert, Fairclough, Van Dijk, to name but a few) tend to see culture (in the sense of beliefs, ideologies etc.) as discursively produced. This perspective has its roots in the work of scholars from Foucault to Bakhtin and Goffman, amongst others. This means that discourse analysis enables us to examine the ways in which discursive practices shape and reshape culture, by paying close attention to the production, circulation and uptake of discourse, as well as the interplay between micro and macro sites of production (Bakhtin’s work on dialogism is particularly useful here, as well as more contemporary notions of indexicality e.g. Silverstein). This approach is based on the assumption that micro and macro contexts are constitutive of one another —some (e.g. Agha) would even go as far as to argue that the micro/macro distinction may be attributed to differences in epistemic traditions, rather than a clearly identifiable boundary between the two. In a similar vein, it’s worth considering what is meant by “culture”. Adrian Holliday’s work on small cultures is a useful starting point for this. Shameless plug: for a more in-depth discussion of these ideas, see my paper: “Myths”, “truths” and the role of Applied Linguistics in contemporary society.

Cultural discourse analysis (CuDA) is an approach to communication which explores culturally distinctive communication practices as these occur in their everyday contexts, the meanings participants activate in those practices, as well as cross-cultural analyses of those everyday practices and their meanings.